Annales Universitatis Paedagogicae Cracoviensis. Folia 206. Studia

16 •2017

206 Annales Universitatis Paedagogicae Cracoviensis

ISSN 2081-545X Studia Mathematica

16 •2017 Editorial Team Wojciech Chachólski, Department of Mathematics, KTH, Sweden M ärton Elekes, Alfred Reny i Institute of Mathematics, Hungarian Academy of Sciences, Hungary Alex Küronya, Institut für Mathematik, Goethe-Universität Frankfurt, Ger many Mohammad Sal Moslehian, Department of Pure Mathematics, Ferdowsi Uni versity of Mashhad, Iran Zsolt Päles, Institute of Mathematics, University of Debrecen, Hungary Tomasz Szemberg, Institute of Mathematics, Pedagogical University of Kraków, Poland Justyna Szpond, (Managing Editor) - Institute of Mathematics, Pedagogical University of Kraków, Poland Behrouz Taji, Department of Mathematics, Albert Ludwigs University of Freiburg, Germany

Beata Hejmej, (Secretary) - Institute of Mathematics, Pedagogical University of Krakow, Poland Paweł Solarz, (Technical Editor) - Institute of Mathematics, Pedagogical Uni versity of Kraków, Poland Magdalena Piszczek, (Technical Editor) - Institute of Mathematics, Pedagogi cal University of Kraków, Poland

Contact st udmat h@up. kr akow. pi

Available online at http: / / studmath.up.krakow.pl http: / / www.degruyter.com/view/j / aupcsm Contents

Tomasz Szemberg, Justyna Szpond, Editorial 5

Nanjundan Magesh, Jagadeesan Yamini, Fekete-Szegô inequalities associated with k-th root transformation based on quasi-subordination 7

Marek Karas, Anna Serwatka, Discrete-time market models from the small investor point of view and the first fundamental-type theorem 17

Ioannis K. Argyros, Santhosh George, Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative 41

Joanna Jureczko, Strong sequences and partition relations 51

Akbar Rezaei, Arsham Borumand Saeid, Andrzej Walendziak, Some results on pseudo-Q algebras 61

Raghavendra G. Kulkarni, Intersect a quartic to extract its roots 73

Sk. Nazmul, On type-2 m-topological spaces 77

Edward Tutaj, LikeN ’s - a point of view on natural numbers 95

Zenon Moszner, Translation equation and the Jordan non-measurable continuous functions 117

Report of Meeting, 17th International Conference on Functional Equations and Inequalities, Bçdlewo, Poland, July 9-15, 2017 121

Editorial

I became the managing editor of Annales Universitatis Paedagogicae Cra- coviensis Studia Mathematica in 2006. Running the journal in the open access model appeared a little revolutionary at this time. Next years showed that this was a right decision which attracted new authors and signiﬁcantly improved the journal visibility and impact. In 2009 the journal appeared for the ﬁrst time on the list B of the Polish Ministry of Science and Higher Education. Articles started to be reviewed in the American Mathematical Society service Mathematical Reviews and the European Mathematical Society service zbMATH. The international editorial board was established and expanded stepwise so that it consists now of 8 active editors. Since 2013 the journal is co-published by de Gruyter. It was accepted in the Directory of Open Access Journals in 2013. In 2016 the journal was awarded the DOAJ Seal. It has been a valuable achievement to conclude the 10 years period of service. I take this opportunity to thank my colleagues and collaborators. I owe a lot to Władysław Wilk, who introduced me to the editorial job and made the online model possible. I thank Paweł Solarz who served as the Editorial Secre- tary and Magdalena Piszczek, who took care of typesetting. Many thanks go to Wydawnictwo Naukowe UP and de Gruyter for constant support in publishing the journal timely. While remaining part of the editorial team, I am very glad to pass over the lead to the next generation.

Kraków, 2017 Tomasz Szemberg

It is a great honour, a privilege and an obligation to take over the Managing Editor post. As an author, I know how important are quick, accurate and mean- ingful reports from the referees. As an editor I will put great eﬀort in treating the manuscripts timely and fairly and to accept those recommended manuscripts which have potential to serve the international mathematical community. In 2017 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica has been accepted to Clarivate Analytics - Emerging Sources Citation Index. This has already resulted in an increase of submissions, which in turn allows for more selective procedures. The editorial team has been joined by Beata Hejmej, who in an energetic manner takes care of fulﬁlling the two main objectives mentioned above. With my fellow editors I am anxious to welcome new authors and high quality manuscripts in the months and years to come.

Kraków, 2017 Justyna Szpond

Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 7-15 DOI: 10.1515/aupcsm-2017-0001

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Nanjundan Magesh and Jagadeesan Yamini Fekete-Szegö inequalities associated with kth root transformation based on quasi-subordination Communicated by Tomasz Szemberg

Abstract. Recently, Haji Mohd and Darus [1] revived the study of coefficient problems for univalent functions associated with quasi-subordination. In- spired largely by this article, we provide coefficient estimates with k-th root transform for certain subclasses of S defined by quasi-subordination.

1. Introduction

Denote by A the class of all analytic functions of the type ∞ X n f(z) = z + anz (z ∈ U), (1) n=2 where U = {z ∈ C : |z| < 1}. Also denote by S the class of all analytic univalent functions of the form (1) in U. Let k be a positive integer. A domain D is said to 2π be k-fold symmetric if a rotation of D about the origin through an angle k carries 2πi 2πi D to itself. A function f is said to be k-fold symmetric in U, if f(e k z) = e k f(z) for every z ∈ U. If f is regular and k-fold symmetric in U, then k+1 2k+1 f(z) = b1z + bk+1z + b2k+1z + .... (2) Conversely, if f is given by (2), then f is k-fold symmetric inside the circle of convergence of the series. For f ∈ S given by (1), the kth root transformation is deﬁned by k 1 k+1 2k+1 F (z) = [f(z )] k = z + bk+1z + b2k+1z + .... (3) AMS (2010) Subject Classiﬁcation: 30C45, 30C50. Keywords and phrases: Univalent functions, starlike functions, convex functions, subordination, quasi-subordination. Corresponding author N. Magesh ([email protected]). [8] Nanjundan Magesh and Jagadeesan Yamini

For two analytic functions f and g, the function f is quasi-subordinate to g in the open unit disc U, if there exist analytic functions h and w, with |h(z)| ≤ 1, f(z) w(0) = 0 and |w(z)| < 1, such that h(z) is analytic in U and written as f(z) ≺ g(z)(z ∈ ) h(z) U and it is denoted by f(z) ≺q g(z)(z ∈ U) and equivalently f(z) = h(z)g(w(z)) (z ∈ U). It is interesting to note that if h(z) ≡ 1, then f(z) = g(w(z)), so that f(z) ≺ g(z) in U, where ≺ is a subordination between f and g in U. Also notice that if w(z) = z, then f(z) = h(z)g(z) and it is said that f is majorized by g and written as f(z) g(z) in U (see [2]). Let ϕ be an analytic and univalent function with positive real part in U, ϕ(0) = 1, ϕ0(0) > 0 and let ϕ map the unit disk U onto a region starlike with respect to 1 and symmetric with respect to the real axis. The Taylor’s series expansion of such a function is

2 3 ϕ(z) = 1 + B1z + B2z + B3z + ..., (4) where all coeﬃcients are real and B1 > 0. Recently, El-Ashwah and Kanas [3] introduced and studied the following two subclasses: n 1 zf 0(z) o S∗(γ, ϕ) := f ∈ A : − 1 ≺ ϕ(z) − 1, z ∈ , γ ∈ \{0} q γ f(z) q U C and n 1 zf 00(z) o K (γ, ϕ) := f ∈ A : ≺ ϕ(z) − 1, z ∈ , γ ∈ \{0} . q γ f 0(z) q U C

∗ We note that, when h(z) ≡ 1, the classes Sq (γ, ϕ) and Kq(γ, ϕ) reduce respectively, to the familiar classes S∗(γ, ϕ) and K(γ, ϕ) of Ma-Minda starlike and convex func- ∗ tions of complex order γ (γ ∈ C\{0}) in U (see [4]). For γ = 1, the classes Sq (γ, ϕ) ∗ and Kq(γ, ϕ) reduce to the classes Sq (ϕ) and Kq(ϕ) studied by Haji Mohd and ∗ Darus [1]. When h(z) ≡ 1, the classes Sq (ϕ) and Kq(ϕ) reduce respectively, to well known subclasses S∗(ϕ) and K(ϕ) introduced and studied by Ma and Minda [5]. By specializing 1 + (1 − 2α)z ϕ(z) = (0 ≤ α < 1) 1 − z or 1 + z β ϕ(z) = (0 < β ≤ 1) 1 − z the classes S∗(ϕ) and K(ϕ) consist of functions known as the starlike (respectively convex) functions of order α or strongly starlike (respectively convex) functions of order β, respectively. kth-root transformation based on quasi-subordination [9]

δ,λ A function f ∈ A given by (1) is said to be in the class Mq (γ, ϕ), 0 6= γ ∈ C, δ ≥ 0, if the following quasi-subordination condition is satisﬁed

0 00 1 zFλ(z) zFλ (z) (1 − δ) + δ 1 + 0 − 1 ≺q ϕ(z) − 1 (z ∈ U), γ Fλ(z) Fλ(z) where 0 Fλ(z) = (1 − λ)f(z) + λzf (z) (0 ≤ λ ≤ 1). We note that,

δ,0 δ 1. Mq (γ, ϕ) := Mq(γ, ϕ),

δ δ 2. Mq(1, ϕ) := Mq(ϕ), [1, Definition 1.7, p.3], 0,0 ∗ 3. Mq (γ, ϕ) := Sq (γ, ϕ), [3, Definition 1.1, p.680], ∗ ∗ 4. Sq (1, ϕ) := Sq (ϕ), [1, Definition 1.1, p.2], 1,0 5. Mq (γ, ϕ) := Kq(γ, ϕ), [3, Definition 1.3, p.681],

6. Kq(1, ϕ) := Kq(ϕ), [1, Deﬁnition 1.3, p.2],

7. For 0 6= γ ∈ C, 0 ≤ λ ≤ 1,

0,λ Mq (γ, ϕ) ≡ Pq(γ, λ, ϕ) n 1 zf 0(z) + λz2f 00(z) o = f ∈ A : − 1 ≺ ϕ(z) − 1, z ∈ . γ (1 − λ)f(z) + λzf 0(z) q U

8. For 0 6= γ ∈ C, 0 ≤ λ ≤ 1,

1,λ Mq (γ, ϕ)

≡ Kq(γ, λ, ϕ) n 1 zf 0(z) + (1 + 2λ)z2f 00(z) + λz3f 000(z) = f ∈ A : − 1 ≺ ϕ(z) − 1, γ zf 0(z) + λz2f 00(z) q o z ∈ U .

Inspired by the papers of [1, 3, 6, 7, 8], we obtain the upper bounds |bk+1| and δ,λ |b2k+1| for f ∈ Mq (γ, ϕ). Also, we investigate the Fekete-Szegö results for the δ,λ class Mq (γ, ϕ) and its special cases. In order to discuss our results we provide the following lemmas.

Lemma 1.1 ([9]) Let w be an analytic function with w(0) = 0, |w(z)| < 1 and let

2 w(z) = u1z + u2z + ... (z ∈ U). (5)

Then for t ∈ C, 2 |u2 − tu1| ≤ max[1; |t|]. [10] Nanjundan Magesh and Jagadeesan Yamini

Lemma 1.2 ([9]) Let h be an analytic function with |h(z)| < 1 and let

2 h(z) = h0 + h1z + h2z + ... (z ∈ U). (6) Then 2 |h0| ≤ 1 and |hn| ≤ 1 − |h0| ≤ 1 (n > 0). Lemma 1.3 ([10]) Let w be the analytic function with w(0) = 0, |w(z)| < 1 and given by (5). Then |w1| ≤ 1 and for any integer n ≥ 2,

2 |un| ≤ 1 − |u1| .

2. Main result

Unless otherwise stated, throughout the sequel, we set f is of the form (1) and ϕ, h and w are given by (4), (6) and (5), respectively. δ,λ In the following theorem, we ﬁnd Fekete-Szgeö result for f ∈ Mq (γ, ϕ). Theorem 2.1 δ,λ Let f ∈ Mq (γ, ϕ) and let F be given by (3). Then |γ|B |b | ≤ 1 , k+1 k(1 + δ)(1 + λ)

γ(1+3δ)k+(1−k)γ(1+2δ)(1+2λ) 2 |γ|{B1 + max{B1, | 2 2 |B1 + |B2|}} |b | ≤ k(1+δ) (1+λ) 2k+1 2k(1 + 2δ)(1 + 2λ) and for µ ∈ C,

γ(1+3δ)k+(1−2µ−k)γ(1+2δ)(1+2λ) 2 |γ|{B1 + max{B1, | 2 2 |B1 + |B2|}} |b − µb2 | ≤ k(1+δ) (1+λ) . 2k+1 k+1 2k(1 + 2δ)(1 + 2λ)

δ,λ Proof. Since f ∈ Mq (γ, ϕ), there exist ϕ and w with |ϕ(z)| ≤ 1, w(0) = 0 and |w(z)| < 1 such that 0 00 1 zFλ(z) zFλ (z) (1 − δ) + δ 1 + 0 − 1 = h(z)(ϕ(w(z)) − 1) (7) γ Fλ(z) Fλ(z) and

2 2 h(z)(ϕ(w(z)) − 1) = h0B1u1z + [h1B1u1 + h0(B1u2 + B2u1)]z + .... (8) From (7) and (8) we get 1 (1 + δ)(1 + λ)a = h B u (9) γ 2 0 1 1 kth-root transformation based on quasi-subordination [11] and 1 2(1 + 2δ)(1 + 2λ)a − (1 + 3δ)(1 + λ)2a2 = h B u + h B u + h B u2. (10) γ 3 2 1 1 1 0 1 2 0 2 1

Equation (9) yields γh B u a = 0 1 1 . (11) 2 (1 + δ)(1 + λ) By subtracting (10) from (9) and using (11) we obtain

γ h γh2B2(1 + 3δ) i a = h B u + h B u + h B + 0 1 u2 . (12) 3 2(1 + 2δ)(1 + 2λ) 1 1 1 0 1 2 0 2 (1 + δ)2 1

For a given f ∈ S of the form (1), we deﬁne F by

k 1 F (z) = [f(z )] k a ha k − 1 i = z + 2 zk+1 + 3 − a2 z2k+1 + ... k k 2k2 2 k+1 2k+1 = z + bk+1z + b2k+1z + ..., where a a k − 1 b = 2 , b = 3 − a2 and so on. (13) k+1 k 2k+1 k 2k2 2 It follows from (11), (12) and (13) that

a γh B u b = 2 = 0 1 1 k+1 k k(1 + δ)(1 + λ) and a k − 1 b = 3 − a2 2k+1 k 2k2 2 2 2 γh0B1 (1+3δ) 2 γ[h1B1u1 + h0B1u2 + (h0B2 + 2 )u1] = (1+δ) 2k(1 + 2δ)(1 + 2λ) k − 1 γ2h2B2u2 − 0 1 1 . 2k2 (1 + δ)2(1 + λ)2

For µ ∈ C we get

2 b2k+1 − µbk+1

γB1 n hB2 γh0B1(1 + 3δ) = h1u1 + h0 u2 + + 2 2k(1 + 2δ)(1 + 2λ) B1 (1 + δ) γh B (1 + 2δ)(1 + 2λ) γh B (1 − 2µ)(1 + 2δ)(1 + 2λ)i o − 0 1 + 0 1 u2 . (1 + δ)2(1 + λ)2 k(1 + δ)2(1 + λ)2 1

Since h is analytic and bounded in U we have

2 |hn| ≤ 1 − |h0| ≤ 1 (n > 0). [12] Nanjundan Magesh and Jagadeesan Yamini

By using this fact and the well-known inequality

|u1| ≤ 1, from Lemma 1.3, we conclude that |γ|B |b | ≤ 1 k+1 k(1 + δ)(1 + λ) and

2 |b2k+1 − µbk+1|

|γ|B1 n h−B2 ≤ 1 + u2 − 2k(1 + 2δ)(1 + 2λ) B1 γ(1 + 3δ)k − γ(1 + 2δ)(1 + 2λ)k + γ(1 − 2µ)(1 + 2δ)(1 + 2λ) i o − h B u2 . k(1 + δ)2(1 + λ)2 0 1 1 In view of Lemma 1.1, we have

γ(1+3δ)k+(1−2µ−k)γ(1+2δ)(1+2λ) 2 |γ|{B1 + max{B1, | 2 2 |B1 + |B2|}} |b − µb2 | ≤ k(1+δ) (1+λ) . 2k+1 k+1 2k(1 + 2δ)(1 + 2λ) When µ = 0, we obtain

γ(1+3δ)k+(1−k)γ(1+2δ)(1+2λ) 2 |γ|{B1 + max{B1, | 2 2 |B1 + |B2|}} |b | ≤ k(1+δ) (1+λ) . 2k+1 2k(1 + 2δ)(1 + 2λ) Hence we obtained the required inequalities of Theorem 2.1.

3. Concluding remarks and corollaries

δ,λ In light of the special subclasses of the class Mq (γ, ϕ), we have the following corollaries and remarks.

Remark 3.1 For δ = λ = 0 and γ = 1, Theorem 2.1 reduces to [6, Theorem 2.1, p.619]. For δ = λ = 0 and γ = k = 1, Theorem 2.1 reduces to [1, Theorem 2.1, p.4].

Corollary 3.2 If f ∈ Kq(γ, ϕ), then |γ|B |b | ≤ 1 , k+1 2k |γ|h n |γ|(k + 3) oi |b | ≤ B + max B , B2 + |B | 2k+1 6k 1 1 4k 1 2 and for µ ∈ C, |γ|h n |γ||k + 3(1 − 2µ)| oi |b − µb2 | ≤ B + max B , B2 + |B | . 2k+1 k+1 6k 1 1 4k 1 2 kth-root transformation based on quasi-subordination [13]

Remark 3.3 For γ = k = 1, Corollary 3.2 reduces to [1, Theorem 2.4, p.7].

Remark 3.4 Taking λ = 0 and γ = 1, Theorem 2.1 coincides with [6, Theorem 2.2, p.620]. Also, for λ = 0 and γ = k = 1, Theorem 2.1 reduces to [1, Theorem 2.10, p.10].

Corollary 3.5 If f ∈ Pq(γ, λ, ϕ), then

|γ|B |b | ≤ 1 , k+1 k(1 + λ) |γ| h n |γ||1 + (1 − k)2λ| oi |b | ≤ B + max B , B2 + |B | 2k+1 2k(1 + 2λ) 1 1 k(1 + λ)2 1 2 and for µ ∈ C,

2 |b2k+1 − µbk+1| |γ| h n |(1 − 2µ)(1 + 2λ) − 2kλ| oi ≤ B + max B , |γ|B2 + |B | . 2k(1 + 2λ) 1 1 k(1 + λ)2 1 2

Corollary 3.6 If f ∈ Kq(γ, λ, ϕ), then

|γ|B |b | ≤ 1 , k+1 2k(1 + λ) |γ| h n |γ||3(1 + 2λ) + k(1 − 6λ)| oi |b | ≤ B + max B , B2 + |B | 2k+1 6k(1 + 2λ) 1 1 4k(1 + λ)2 1 2 and for µ ∈ C,

2 |b2k+1 − µbk+1| |γ| h n |3(1 − 2µ)(1 + 2λ) + k(1 − 6λ)| oi ≤ B + max B , |γ|B2 + |B | . 6k(1 + 2λ) 1 1 4k(1 + λ)2 1 2

Remark 3.7 For γ = 1 and k = 1, Corollary 3.6 corrects the results in [8, Theorem 2.1, p.195].

Remark 3.8 For k = 1, the results discussed in present paper coincide with the results obtained in [11].

Acknowledgement. The author would like to thank the editor and the anonymous referees for their comments and suggestions on this paper. Further, the present investigation of the second-named author was supported by UGC under the grant F.MRP-1397 /14-15 /KABA084 /UGC-SWRO. [14] Nanjundan Magesh and Jagadeesan Yamini

References

[1] Haji Mohd, Maisarah, and Maslina Darus. "Fekete-Szegő problems for quasi- subordination classes." Abstr. Appl. Anal. 2012: Art. ID 192956, 14 pp. Cited on 7, 8, 9, 12 and 13. [2] Robertson, Malcolm S. "Quasi-subordination and coefficient conjectures." Bull. Amer. Math. Soc. 76 (1970): 1–9. Cited on 8. [3] El-Ashwah, Rabha, and Stanisława Kanas. "Fekete-Szegö inequalities for quasi- subordination functions classes of complex order." Kyungpook Math. J. 55, no. 3 (2015): 679–688. Cited on 8 and 9. [4] Ravichandran, V., Yasar Polatoglu, Metin Bolcal, and Arzu Sen. "Certain subclasses of starlike and convex functions of complex order." Hacet. J. Math. Stat. 34 (2005): 9–15. Cited on 8. [5] Ma, Wan Cang, and David Minda. "A unified treatment of some special classes of univalent functions." In Proceedings of the Conference on Complex Analysis (Tianjin, 1992). Vol I of Conf. Proc. Lecture Notes Anal., 157–169. Cambridge, MA: Int. Press, 1994. Cited on 8. [6] Gurusamy, Palpandy, and Janusz Sokół, and Srikandan Sivasubramanian. "The Fekete-Szegö functional associated with k-th root transformation using quasi- subordination." C. R. Math. Acad. Sci. Paris 353, no. 7 (2015): 617–622. Cited on 9, 12 and 13. [7] Goyal, Som Prakash, and Onkar Singh. "Fekete-Szegö problems and coefficient estimates of quasi-subordination classes." J. Rajasthan Acad. Phys. Sci. 13, no. 2 (2014): 133–142. Cited on 9. [8] Keerthi, Bhaskara Srutha, and P. Lokesh. "Fekete-Szegö problem for certain sub- class of analytic univalent function using quasi-subordination." Math. AEterna 3, no. 3-4 (2013): 193–199. Cited on 9 and 13. [9] Keogh, F.R., and Edward P. Merkes. "A coefficient inequality for certain classes of analytic functions." Proc. Amer. Math. Soc. 20 (1969): 8–12. Cited on 9 and 10. [10] Duren, Peter L. Univalent functions. Vol. 259 of Grundlehren der Mathematischen Wissenschaften. New York: Springer-Verlag, 1983. Cited on 10. [11] Magesh, Nanjundan, and V.K. Balaji, and C. Abirami. "Fekete-Szegö inequalities for certain subclasses of starlike and convex functions of complex order associated with quasi-subordination." Khayyam J. Math. 2, no. 2 (2016): 112–119. Cited on 13.

Nanjundan Magesh Post-Graduate and Research Department of Mathematics Government Arts College for Men Krishnagiri 635001 Tamilnadu India E-mail: [email protected] kth-root transformation based on quasi-subordination [15]

Jagadeesan Yamini Department of Mathematics Government First Grade College Vijayanagar, Bangalore-560104 Karnataka India E-mail: [email protected]

Received: January 25, 2017; ﬁnal version: April 6, 2017; available online: May 26, 2017.

Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 17-40 DOI: 10.1515/aupcsm-2017-0002

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Marek Karaś and Anna Serwatka Discrete-time market models from the small investor point of view and the ﬁrst fundamental-type theorem Communicated by Tomasz Szemberg

Abstract. In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing. For the standard approach a risk-free bank account process is used as numeraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the model of a market (with d risky assets), which does not hold the same interest rate assumptions. We introduce two predictable processes for modelling deposits and loans. We propose a new concept of a martingale pair for the market and prove that if there exists a martingale pair for the considered market, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free.

1. Introduction

In this paper, we will discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing or the Dalang-Morton- Willinger theorem [1]. It states that for the standard discrete-time, finite horizon market model there is no arbitrage opportunity if and only if the price process is a martingale, with respect to an equivalent probability measure. There are various

AMS (2010) Subject Classification: 14Rxx, 14R10. Keywords and phrases: market model, arbitrage strategy, arbitrage opportunity, arbitrege- free market. [18] Marek Karaś and Anna Serwatka proofs of this theorem in existence, which use different areas of mathematics, for more detail see [1, 6, 3]. The Fundamental Theorem of Asset Pricing for a model with finite Ω was proven by M. Harrison and D. Kreps in 1979 [4]. Harrison and Pliska [5] proved a more general version of this theorem. In [2] Delbaen and Schachermayer show a concept which characterizes the existence of an equivalent martingale measure for a general class of processes in terms of the no free lunch with vanishing risk. The main theorem of that paper is the general version of the Dalang-Morton-Willinger theorem for real valued semi-martingales. The Fundamental Theorem of Asset Pricing was studied in detail by many mathematicians, who checked various aspects and focused on additional equivalent conditions of this theorem. Many new theorems were proved through the investigation of different aspects of the Dalang-Morton-Willinger theorem. Among them are theorems which include taxes like in [7], where the author gives the necessary and sufficient conditions for a linear taxation system to be neutral-within the multi-period discrete time in a no arbitrage model. Kabanov and Safarian worked on multi-asset discrete-time models with friction and gave in [10] conditions equivalent to the absence of arbitrage in markets with friction. The theory goes further and there are papers with equivalent conditions for the absence of so-called weak arbitrage [9] and robust no-arbitrage opportunities [12]. It is also an interesting concept to consider the models with bid and ask price processes in [8] and [11]. Rola in [11] considers a market with a multi-dimensional bid and ask processes and with a money account, introduces the notion of an equivalent bid-ask martingale measure and proves that the existence of such a measure is equivalent to no-arbitrage in this model. In many papers on arbitrage in discrete market models the authors consider models containing d+1 financial assets: one risk-free asset {Bt}t=0,1,...,T (which is i interpreted as a bank account) and d risky assets {St}t=0,1,...,T for i ∈ {1, . . . , d} (say i.e. stocks). For the standard approach the risk-free bank account process is used as numéraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the model of a market (with d risky assets), which does not hold the same interest rate assump- + − tions. We introduce two predictable processes {Bt }t=0,1,...,T and {Bt }t=0,1,...,T for modelling deposits and loans. We propose a new concept of a martingale pair ∗ ({Bt}t=0,...,T ,P ) for the market M = (S, P) and prove that if a martingale pair for the considered market exists, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free.

2. Model description

We assume that there is a given probability space (Ω, F,P ), a ﬁnite number T ∈ N+ = N ∪ {0} called the time horizon, and a ﬁltration {Ft}t=0,1,...,T of the measurable space (Ω, F). We propose the following model of a market which does not satisfy the assumption that the interest rates for borrowing and saving money are the same. Discrete-time market models from the small investor point of view [19]

Definition 2.1 By the model of a market (with d risky assets), which does not hold the same interest rates assumption, we mean, in general, the triple M = (S, P, ϕ), where + − 1 d • S = {(Bt ,Bt ,St ,...,St )}t=0,1,...,T is the adapted stochastic process, with d+2 + − values in (0, +∞) , such that the processes {Bt }t=0,1,...,T and {Bt }t=0,1,...,T are predictable, and

+ − B0 = B0 = 1, + − Bt+1 Bt+1 + ≤ − for all t ∈ {0,...,T − 1}, Bt Bt

•P is a subset of the set of all predictable stochastic processes Θ = + − 1 d 2 d {Θt}t=0,...,T = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T , with values in [0, +∞) × R , 1 d • ϕ is the adapted process {(ϕt , . . . , ϕt )}t=0,1,...,T , with values in the space ([0, +∞)R)d (i.e. for each t ∈ {0,...,T }, each i ∈ {0, . . . , d} and each ω ∈ Ω, i ϕt(ω) is a function R → [0, +∞)). + We assume that the process {Bt }t=0,1,...,T is modelling the changes, in time, of the value of one unit of money given in the bank deposit at time t = 0, and − is called deposit process, while the process {Bt }t=0,1,...,T , called loan process, is modelling the amount of money that must be given back to the bank at time t, if one has borrowed one unit of money from the bank at time t = 0. i We also assume, as usual, that the processes {St}t=0,1,...,T , for i = 1, . . . , d, are modelling the prices of the risky assets, say stocks.

Definition 2.2 Assume that the market M = (S, P, ϕ) is given. By a portfolio on the market M we mean any vector Θ = (Θ+, Θ−, Θ1,..., Θd) ∈ [0, +∞)2 × Rd. The value of the portfolio Θ = (Θ+, Θ−, Θ1,..., Θd) at time t is deﬁned by

Θ + + − − 1 1 d d Vt = Θ Bt − Θ Bt + Θ St + ... + Θ St .

Definition 2.3 By the strategy (or trading strategy) on the market M we mean any predictable + − 1 d process Θ = {Θt}t=0,...,T = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T with values in [0, +∞)2 × Rd, which is an element of the set P, called the set of all strategies. In our consideration, we will assume that P consists of all predictable processes + − 1 d 2 Θ = {Θt}t=0,...,T = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T , with values in [0, +∞) × Rd, but if one wants to consider the market with some additional restriction on strategies, then P will not consist of all such processes. Since for t = 0 we must know what F−1 means, we assume that F−1 = {∅, Ω}. For the simplicity of the considerations we will also assume that F0 = {∅, Ω}. 2 One can see that for any ω ∈ Ω and for any t ∈ {0,...,T }, Θt(ω) ∈ [0, +∞) × Rd is a portfolio, and we assume that this portfolio is held from time t − 1 up to time t. This justiﬁes our assumption that the process {Θt}t=0,...,T is predictable. [20] Marek Karaś and Anna Serwatka

Let us note that for any given strategy {Θt}t=0,...,T on the market M = (S, P, ϕ), the value of the strategy at time t can be calculated in two ways, namely as the sum + + − − 1 1 d d Θt Bt − Θt Bt + Θt St + ... + Θt St or as the sum + + − − 1 1 d d Θt+1Bt − Θt+1Bt + Θt+1St + ... + Θt+1St .

This is because the portfolio Θt is held during the time interval (t − 1, t), while the portfolio Θt+1 is held during the time interval (t, t + 1). Thus, for any t ∈ {0,...,T −1}, we consider two possible different values of the strategy {Θt}t=0,...,T at time t, called the value before transaction and the value after transaction. Definition 2.4 Assume that we are given a strategy {Θt}t=0,...,T defined on the market M = (S, P, ϕ). Then, the value of the strategy {Θt}t=0,...,T at time t before transaction, we define as Θ + + − − 1 1 d d Vt− := Θt Bt − Θt Bt + Θt St + ... + Θt St and the value of the strategy {Θt}t=0,...,T at time t after transaction, we define as Θ + + − − 1 1 d d Vt+ := Θt+1Bt − Θt+1Bt + Θt+1St + ... + Θt+1St . For the terminal date T , we can only consider the value before transaction Θ + + − − 1 1 d d VT − := ΘT BT − ΘT BT + ΘT ST + ... + ΘT ST . Θ One can notice that, for a given strategy {Θt}t=0,...,T , the values Vt− and Θ Θ Θ Vt+ can be different. The inequality Vt+(ω) > Vt−(ω), for some ω ∈ Ω, means that we add an amount of money to the system/strategy, while the inequality Θ Θ Vt+(ω) < Vt−(ω), for some ω ∈ Ω, can mean that we subtract an amount of money from the system/strategy. Definition 2.5 1 Let us assume that the market M = (S, P, ϕ) is given. The process ϕ = {(ϕt ,..., d ϕt )}t=0,1,...,T is called the transaction cost process, and is assumed that for each i ∈ {1, . . . , d}, each ω ∈ Ω, each t ∈ {0, 1,...,T } and each x ∈ R, the value i i of the function ϕt(ω): R → [0, +∞) at x, denoted by ϕt(ω).x, is equal to the transaction cost of buying x shares of i-th stock at time t if x > 0, and is equal to the transaction cost of selling |x| shares of i-th stock at time t if x < 0. For a given strategy {Θt}t=0,...,T defined on the market M = (S, P, ϕ), the total transaction cost at time t ∈ {0, 1,...,T } is defined as

d d Θ X i i i X i i Ct = ϕt.(Θt+1 − Θt) =: ϕt.∆Θt, i=1 i=1 i i where for any function ζ :Ω → R, by ϕt.ζ we denote the function ω 7→ ϕt(ω).ζ(ω). For the terminal date T , when (by assumption) all shares are sold, the total transaction cost is deﬁned as d Θ X i i CT = ϕT .(−ΘT ). i=1 Discrete-time market models from the small investor point of view [21]

The following condition means that the market M is assumed to be without transaction costs

i ∀ i ∈ {1, . . . , d}, t ∈ {0, 1,...,T }, ω ∈ Ω ϕt(ω) ≡ 0.

In case the market M is without transaction costs, we will write M = (S, P) instead of M = (S, P, ϕ). On the other hand, if for each i ∈ {1, . . . , d}, each t ∈ {0, 1,...,T } and each i i ω ∈ Ω, the functions ϕt(ω)|[0,+∞) and ϕt(ω)|(−∞,0]) are linear, we say that M is a market with proportional transaction costs. Usually, it is assumed that the i functions ϕt(ω) do not depend on t and ω. Now we can deﬁne the value of the strategy at terminal time T , after transac- tions.

Definition 2.6 For a given strategy {Θt}t=0,...,T deﬁned on the market M, we set (by deﬁnition)

Θ Θ Θ VT + = VT − − CT .

Θ The value VT + will be called terminal value of the strategy Θ and also denoted Θ by VT . Definition 2.7 The strategy {Θt}t=0,...,T (deﬁned on the market M = (S, P, ϕ)) is called a self- ﬁnancing strategy if

Θ Θ Θ Vt+ = Vt− − Ct for all t ∈ {0,...,T }. (1)

If the market M is without transaction costs, the above condition simplifies as follows Θ Θ Vt+ = Vt− for all t ∈ {0,...,T }. (2) Let us notice that the condition (1) or (2) must be checked only for t ∈ Θ {0,...,T − 1}, because for t = T the condition is valid by the definition of VT +. Thus for a self-financing strategy {Θt}t=0,...,T defined on the market without Θ Θ transaction costs, the common value Vt+ = Vt−, for t ∈ {0,...,T }, is called the Θ Θ value of the strategy at time t and denoted by Vt . In particular, V0 is called the initial value of the strategy. In a general situation, by the initial value of the strategy {Θt}t=0,...,T , we Θ Θ mean V0−, which can also be denoted by V0 . Definition 2.8 A self-financing strategy {Θt}t=0,...,T is called an arbitrage opportunity or arbitrage Θ strategy if V0 = 0 and the terminal value of the strategy satisfies

Θ Θ P (VT ≥ 0) = 1 and P (VT > 0) > 0.

In the sequel we will assume that the market M is without transaction costs. The following lemma is an easy but useful observation. [22] Marek Karaś and Anna Serwatka

Lemma 2.9 If there is a trading strategy {Θt}t=0,...,T , deﬁned on the market M = (S, P), such Θ Θ Θ Θ Θ that V0− = 0, Vt− ≥ Vt+ for t ∈ {0,...,T }, and P (VT + ≥ 0) = 1, P (VT + > 0) > 0, then there is an arbitrage opportunity on the market M.

+ − 1 d Proof. Let’s create a strategy {Φt}t=0,...,T = {(Φt , Φt , Φt ,..., Φt )}t=0,...,T in the following way + − 1 d Φ0 = (Θ0 , Θ0 , Θ0,..., Θ0) and + − 1 d Φt+1 = (Φt+1, Θt+1, Θt+1,..., Θt+1) for t = 0,...,T − 1, where

Θ Θ + + + Vt− − Vt+ + Φt+1 = Φt − Θt + + + Θt+1. Bt

We have proved that the process {Φt}t=0,...,T is indeed a strategy and then that it is an arbitrage opportunity on the market M. + To see that {Φt}t=0,...,T ∈ P, we must check that Φt ≥ 0 for all t ∈ {0,...,T }, + + so ﬁrst we verify this by using induction that Φt ≥ Θt for any t ∈ {0,...,T }. + + For t = 0, we have Φ0 = Θ0 . Let t0 ∈ {0,...,T − 1} be given and assume Φ+ ≥ Θ+ . Then, because Φ+ − Θ+ ≥ 0 and V Θ ≥ V Θ , we have Φ+ = t0 t0 t0 t0 t0− t0+ t0+1 V Θ −V Θ Φ+ − Θ+ + t0− t0+ + Θ+ ≥ Θ+ and the proof of the induction step is t0 t0 B+ t0+1 t0+1 t0 completed. Next we check that the strategy {Φt}t=0,...,T is self-ﬁnancing. Note that

Φ + + − − 1 1 d d + + Θ + + Θ Vt− = Φt Bt − Θt Bt + Θt St + ... + Θt St = Φt Bt + Vt− − Θt Bt ≥ Vt−. (3)

Therefore,

Φ + + − − 1 1 d d Vt+ = Φt+1Bt − Θt+1Bt + Θt+1St + ... + Θt+1St Θ Θ + + Vt− − Vt+ + + − − 1 1 = Φt − Θt + + + Θt+1 Bt − Θt+1Bt + Θt+1St Bt d d + ... + Θt+1St + + + + Θ Θ + + − − 1 1 = Φt Bt − Θt Bt + Vt− − Vt+ + Θt+1Bt − Θt+1Bt + Θt+1St d d + ... + Θt+1St + + + + Θ = Φt Bt − Θt Bt + Vt− Φ = Vt−.

Φ Θ Φ Φ Θ Θ Finally, since V0 = V0− = 0 and (3) VT = VT − ≥ VT − = VT +, we have Φ Φ Θ Φ Θ P (V0 = 0) = 1, P (VT ≥ 0) ≥ P (VT ≥ 0) = 1 and P (VT > 0) ≥ P (VT + > 0) > 0. Thus we have proved that the strategy {Φt}t=0,...,T is an arbitrage opportunity on the market M. Discrete-time market models from the small investor point of view [23]

Theorem 2.10 1 d For any predictable process {(Θt ,..., Θt )}t=0,...,T and any number v0 ∈ R there + − 2 is exactly one predictable process {(Θt , Θt )}t=0,...,T with values in [0, +∞) such that + − 1 d (i) the process {(Θt , Θt , Θt ,..., Θt )}t=0,...,T is a self-ﬁnancing strategy with the initial value v0, + − + (ii) for any t ∈ {0,...,T } we have Θt · Θt = 0 (in other words {Θt = 0} ∪ − {Θt = 0} = Ω).

+ − Proof. First, we prove the existence of a predictable process {(Θt , Θt )}t=0,...,T . 1 1 d d 1 1 d d Denote Xt− = Θt St + ... + Θt St and Xt+ = Θt+1St + ... + Θt+1St . Now, for + − 1 d the strategy Θ = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T given by an arbitrary predictable + − 2 Θ + + process {(Θt , Θt )}t=0,...,T with values in [0, +∞) , we can write Vt− = Θt Bt − − − Θ + + − − Θt Bt + Xt− and Vt+ = Θt+1Bt − Θt+1Bt + Xt+. Let us observe that X0− is a constant number and Xt+ is Ft-measurable for + − t = 0,...,T − 1. We will construct a process {(Θt , Θt )}t=0,...,T inductively with respect to t ∈ {0,...,T }. For t = 0, we set ( + − (v0 − X0−, 0) if v0 − X0− ≥ 0, (Θ0 , Θ0 ) := (4) (0,X0− − v0) otherwise, for t ∈ {0,...,T −1}, assuming that we have already constructed Ft−1-measurable + − variables Θt and Θt , we put Θ + Vt− − Xt+ Θ := 1l Θ (5) t+1 + {Vt−−Xt+≥0} Bt and Θ − Xt+ − Vt− Θ := 1l Θ (6) t+1 − {Vt−−Xt+<0}. Bt + − Regardless of the fact that in the equations above a process {(Θt , Θt )}t=0,...,T Θ is not entirely defined, the value of this strategy at time t before the transaction Vt− + − is known because we have assumed that Θt and Θt have already been constructed. + − 1 d Now we may easily check that the process Θ = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T defined by (4)–(6) is predictable and satisfies conditions of the theorem. First notice that, by (4), we have

Θ + + − − + − V0− = Θ0 B0 − Θ0 B0 + X0− = Θ0 − Θ0 + X0− = v0. On the other hand, by (5) and (6), for any t ∈ {0,...,T − 1} we get

Θ + + − − Vt+ = Θt+1Bt − Θt+1Bt + Xt+ Θ Θ = (V − Xt+)1l Θ − (Xt+ − V )1l Θ + Xt+ t− {Vt−−Xt+≥0} t− {Vt−−Xt+<0} Θ = Vt−. + − To see that Θt , Θt are Ft−1-measurable, for t ∈ {0,...,T }, let us notice, that + − + − Θ0 , Θ0 are constants and that for a given t ∈ {0,...,T − 1} if Θt , Θt are Ft−1- + − measurable, then by (5) and (6) Θt+1, Θt+1 are Ft-measurable. Thus, we have [24] Marek Karaś and Anna Serwatka already checked that the condition (i) is satisfied. The condition (ii) is also satis- + − fied because of the definition of Θt and Θt (see equations (4)–(6). + − Now we prove the uniqueness of the process {(Θt , Θt )}t=0,...,T . Suppose, + − + − conversely, that we have two processes {(Θt , Θt )}t=0,...,T and {(Φt , Φt )}t=0,...,T + − satisfying conditions (i) and (ii) from the theorem. Since Θ0 − Θ0 + X0− = v0 = + − + − + − + − + − Φ0 − Φ0 + X0−, Θ0 , Θ0 , Φ0 , Φ0 ∈ [0, ∞) and Θ0 · Θ0 = 0 = Φ0 · Φ0 it follows + + − − that Θ0 = Φ0 and Θ0 = Φ0 . + + Next, using induction with respect to t ∈ {0,...,T }, we show that Θt = Φt − − and Θt = Φt for all t ∈ {0,...,T }. So, assume that for t0 ∈ {0,...,T − 1}, we have Θ+ = Φ+ and Θ− = Φ− . Then, V Θ = Θ+ B+ − Θ− B− + X = t0 t0 t0 t0 t0− t0 t0 t0 t0 t0− Φ+ B+ − Φ− B− + X = V Φ . Since V Θ = V Θ and V Φ = V Φ , we obtain t0 t0 t0 t0 t0− t0− t0+ t0− t0+ t0− Θ+ B+ − Θ− B− + X = V Θ = V Φ = Φ+ B+ − Φ− B− + X , and t0+1 t0 t0+1 t0 t0+ t0+ t0+ t0+1 t0 t0+1 t0 t0+ thus Θ+ B+ − Θ− B− = Φ+ B+ − Φ− B−. (7) t0+1 t0 t0+1 t0 t0+1 t0 t0+1 t0 Since Φ+ ·Φ− = 0, B+,B− > 0 and Θ+ , Θ− , Φ+ , Φ− ≥ 0, we see by t0+1 t0+1 t0 t0 t0+1 t0+1 t0+1 t0+1 (7), that {Θ− = 0} ⊂ {Φ− = 0} and {Θ+ = 0} ⊂ {Φ+ = 0}. Through t0+1 t0+1 t0+1 t0+1 this symmetry, we also have {Φ− = 0} ⊂ {Θ− = 0} and {Φ+ = 0} ⊂ t0+1 t0+1 t0+1 {Θ+ = 0}, and so {Θ− = 0} = {Φ− = 0} and {Θ+ = 0} = {Φ+ = 0}. t0+1 t0+1 t0+1 t0+1 t0+1 Now, on the set {Θ− = 0} = {Φ− = 0} equation (7) simplifies to t0+1 t0+1

Θ+ B+ = Φ+ B+. (8) t0+1 t0 t0+1 t0

Since B+ > 0, it follows from (8)) that Θ+ = Φ+ on the set {Θ− = 0}. t0 t0+1 t0+1 t0+1 Of course, on the set {Θ+ = 0} = {Φ+ = 0} we also have Θ+ = Φ+ . t0+1 t0+1 t0+1 t0+1 Thus, we get Θ+ = Φ+ , on the set {Θ+ = 0} ∪ {Θ− = 0} = Ω. Similar t0+1 t0+1 t0+1 t0+1 arguments show that Θ− = Φ− on Ω. t0+1 t0+1 Theorem 2.11 1 d Assume that we are given a predictable process {(Θt ,..., Θt )}t=0,...,T . Let + − + − {(Θt , Θt )}t=0,...,T and {(Φt , Φt )}t=0,...,T be two predictable processes such that both processes

+ − 1 d + − 1 d Θ = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T and Φ = {(Φt , Φt , Θt ,..., Θt )}t=0,...,T

Θ Φ are self-ﬁnancing strategies. If V0 ≥ V0 and for all t ∈ {0,...,T − 1} we have + − Θt · Θt = 0, then for all t ∈ {0,...,T },

Θ Φ Vt ≥ Vt . (9)

Moreover, if t0 ∈ {0,...,T − 1} is such that

B+ B− P Φ+ > 0, Φ− > 0, t0+1 < t0+1 > 0, (10) t0+1 t0+1 B+ − t0 Bt0 then for all t ∈ {t0 + 1,...,T },

Θ Φ P (Vt > Vt ) > 0. (11) Discrete-time market models from the small investor point of view [25]

Proof. We prove (9) using induction on t. Let t = 0, then (9) is satisﬁed by the assumption. Assume now that the claim is true for t, where t ∈ {0,...,T − 1}. We show that it is true for t + 1. Since

+ − 1 d + − 1 d Θ = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T and Φ = {(Φt , Φt , Θt ,..., Θt )}t=0,...,T are self-ﬁnancing strategies, we have

Θ Θ + + − − 1 1 d d Vt = Vt+ = Θt+1Bt − Θt+1Bt + Θt+1St + ... + Θt+1St and Φ Φ + + − − 1 1 d d Vt = Vt+ = Φt+1Bt − Φt+1Bt + Θt+1St + ... + Θt+1St . Θ Φ Inequality Vt ≥ Vt implies + + − − + + − − Θt+1Bt − Θt+1Bt ≥ Φt+1Bt − Φt+1Bt . (12)

− + Bt+1 First, consider the situation on the set {Θt+1 = 0}. Multiplying (12) by − and Bt + − Bt+1 Bt+1 using + ≤ − , we obtain Bt Bt − − − + + Bt+1 − − + + − − −Θt+1Bt+1 ≥ Φt+1Bt · − − Φt+1Bt+1 ≥ Φt+1Bt+1 − Φt+1Bt+1. Bt + Therefore, on the set {Θt+1 = 0} we get

Θ Θ − − 1 1 d d Vt+1 = V(t+1)− = −Θt+1Bt+1 + Θt+1St+1 + ... + Θt+1St+1 + + − − 1 1 d d Φ ≥ Φt+1Bt+1 − Φt+1Bt+1 + Θt+1St+1 + ... + Θt+1St+1 = V(t+1)− Φ = Vt+1.

− Consider now the situation on the set {Θt+1 = 0}. Through multiplying (12) by + + − Bt+1 Bt+1 Bt+1 + and using the inequality + ≤ − , we obtain Bt Bt Bt + + + + + − − Bt+1 + + − − Θt+1Bt+1 ≥ Φt+1Bt+1 − Φt+1Bt · + ≥ Φt+1Bt+1 − Φt+1Bt+1. Bt − Hence, also on the set {Θt+1 = 0}, we have

Θ Θ + + 1 1 d d Vt+1 = V(t+1)− = Θt+1Bt+1 + Θt+1St+1 + ... + Θt+1St+1 + + − − 1 1 d d Φ ≥ Φt+1Bt+1 − Φt+1Bt+1 + Θt+1St+1 + ... + Θt+1St+1 = V(t+1)− Φ = Vt+1,

+ − which completes the proof of (9), because {Θt+1 = 0} ∪ {Θt+1 = 0} = Ω. Next we prove (11) by using induction on t. Let us take any

n B+ B− o ω ∈ Φ+ > 0, Φ− > 0, t0+1 < t0+1 . (13) t0+1 t0+1 B+ − t0 Bt0 [26] Marek Karaś and Anna Serwatka

Θ Θ Φ Φ + − i In the sequel, we will write vt for Vt (ω), vt for Vt (ω), θt , θt , θt for, re- + − i + − i + − spectively, Θt (ω), Θt (ω), Θt(ω) and φt , φt , φt for, respectively, Φt (ω), Φt (ω), i + − i Φt(ω). We will use similar notation for Bt (ω), Bt (ω) and St(ω) (there is no possibility of confusion with the cost process, because we consider the market without transaction costs). First we show that V Θ (ω) > V Φ (ω). Since Θ is a self-ﬁnancing strategy, t0+1 t0+1 we have vΘ = V Θ (ω) = θ+ b+ − θ− b− + θ1 s1 + ... + θd sd . t0 t0+ t0+1 t0 t0+1 t0 t0+1 t0 t0+1 t0 Let us set x := θ1 s1 + ... + θd sd , t0 t0+1 t0 t0+1 t0 x := θ1 s1 + ... + θd sd , t0+1 t0+1 t0+1 t0+1 t0+1 additionally vΘ −x = θ+ b+ −θ− b− . Consider the situation when vΘ −x ≥ t0 t0 t0+1 t0 t0+1 t0 t0 t0 0. Then θ− = 0 and vΘ − x = θ+ b+ , so t0+1 t0 t0 t0+1 t0 Θ v − xt θ+ = t0 0 . (14) t0+1 + bt0 Note that vΦ = vΦ = φ+ b+ − φ− b− + θ1 s1 + ... + θd sd t0 t0+ t0+1 t0 t0+1 t0 t0+1 t0 t0+1 t0 (15) = φ+ b+ − φ− b− + x , t0+1 t0 t0+1 t0 t0 therefore Φ − − v − xt + φ b φ+ = t0 0 t0+1 t0 . (16) t0+1 + bt0 One can make the following obvious observation that, because of (13) and the fact that φ− > 0 we have t0+1 b− φ− t0 b+ − φ− b− < 0. (17) t0+1 + t0+1 t0+1 t0+1 bt0 Now, using (9) and (14)–(17), we obtain vΘ = θ+ b+ − θ− b− + θ1 s1 + ... + θd sd t0+1 t0+1 t0+1 t0+1 t0+1 t0+1 t0+1 t0+1 t0+1 Θ Φ v − xt v − xt = θ+ b+ + x = t0 0 b+ + x ≥ t0 0 b+ + x t0+1 t0+1 t0+1 + t0+1 t0+1 + t0+1 t0+1 bt0 bt0 Φ − v − xt b > t0 0 b+ + φ− t0 b+ − φ− b− + x + t0+1 t0+1 + t0+1 t0+1 t0+1 t0+1 bt0 bt0 b+ = (vΦ − x + φ− b− ) t0+1 − φ− b− + x t0 t0 t0+1 t0 + t0+1 t0+1 t0+1 bt0 = φ+ b+ − φ− b− + x t0+1 t0+1 t0+1 t0+1 t0+1 = vΦ . t0+1 Discrete-time market models from the small investor point of view [27]

Further, if vΘ − x < 0, then θ+ = 0 and vΘ − x = −θ− b− , and thus t0 t0 t0+1 t0 t0 t0+1 t0

Θ v − xt −θ− = t0 0 . t0+1 − bt0 As in earlier examples, using (9), (16) and (17), we have

Θ v − xt vΘ = −θ− b− + x = t0 0 b− + x t0+1 t0+1 t0+1 t0+1 − t0+1 t0+1 bt0 vΦ − x vΦ − x t0 t0 − t0 t0 + ≥ b + xt +1 > b + xt +1 − t0+1 0 b+ t0+1 0 bt0 t0 Φ − v − xt b > t0 0 b+ + φ− t0 b+ − φ− b− + x + t0+1 t0+1 + t0+1 t0+1 t0+1 t0+1 bt0 bt0 = φ+ b+ − φ− b− + x t0+1 t0+1 t0+1 t0+1 t0+1 = vΦ . t0+1

Since the afore-mentioned arguments are valid for arbitrary ω ∈ Φ+ > 0, t0+1 B+ B− Φ− > 0, t0+1 < t0+1 , we see that t0+1 B+ B− t0 t0

n B+ B− o Φ+ > 0, Φ− > 0, t0+1 < t0+1 ⊂ {V Θ > V Φ }. t0+1 t0+1 B+ − t0+1 t0+1 t0 Bt0

Θ Φ Assume now that P (Vt > Vt ) > 0 for t, where t ∈ {t0 + 1,...,T − 1}. We show Θ Φ Θ Φ Θ Φ that {Vt > Vt } ⊂ {Vt+1 > Vt+1}. So, let us ﬁx ω ∈ {Vt > Vt } and use the notation which we used in the proof that P (V Θ > V Φ ) > 0. Thus, we have t0+1 t0+1

Θ Θ + + − − vt = vt+ = θt+1bt − θt+1bt + xt.

Θ Θ − + vt −xt Consider the situation when vt − xt ≥ 0. Then θt+1 = 0 and θt+1 = + . On bt the other hand, we have

Φ + + − − vt = φt+1bt − φt+1bt + xt,

vΦ−x +φ− b− Φ + t t t+1 t + − xt−vt so φt+1 = + . Since φt+1 ≥ 0, we see that φt+1 ≥ − . Since bt bt + − bt+1 bt+1 − + ≤ − and φt+1 ≥ 0, we get bt bt

− − bt + − − φt+1 + bt+1 − φt+1bt+1 ≤ 0. bt Therefore, we have

Θ Φ Θ + + vt − xt + vt − xt + vt+1 = θt+1bt+1 + xt+1 = + bt+1 + xt+1 > + bt+1 + xt+1 bt bt [28] Marek Karaś and Anna Serwatka

Φ − vt − xt + − bt + − − ≥ + bt+1 + φt+1 + bt+1 − φt+1bt+1 + xt+1 bt bt + + − − Φ = φt+1bt+1 − φt+1bt+1 + xt+1 = vt+1.

Θ Θ + − vt −xt If vt − xt < 0, then θt+1 = 0 and −θt+1 = − . As in previous cases, we obtain bt

Θ Φ Θ − − vt − xt − vt − xt − vt+1 = −θt+1bt+1 + xt+1 = − bt+1 + xt+1 > − bt+1 + xt+1 bt bt Φ Φ − vt − xt + vt − xt + − bt + − − ≥ + bt+1 + xt+1 ≥ + bt+1 + xt+1 + φt+1 + bt+1 − φt+1bt+1 bt bt bt Φ = vt+1.

Θ Φ Θ Φ Thus, we have proved that {Vt > Vt } ⊂ {Vt+1 > Vt+1} for all t ∈ {t0 + 1,..., T − 1}. This completes the proof of the theorem.

As a consequence of these theorems we obtain the following fact.

Corollary 2.12 If there is an arbitrage strategy on the market M = (S, P), then there is, also, an arbitrage strategy satisfying the condition (ii) of Theorem 2.10.

+ − 1 d Proof. Indeed, if Θ = {(Θt , Θt , Θt ,..., Θt )}t=0,...,T is any arbitrage strategy on the market M = (S, P), then by Theorem 2.10 there is a predictable process + − + − 1 d {(Φt , Φt )}t=0,...,T such that the process Φ = {(Φt , Φt , Θt ,..., Θt )}t=0,...,T is a self-ﬁnancing strategy with the initial value 0. Then, by Theorem 2.11, we have

Φ Θ P (VT ≥ 0) ≥ P (VT ≥ 0) = 1 and Φ Θ P (VT > 0) ≥ P (VT > 0) > 0.

3. Martingale property

+ − Since we have two processes {Bt }t=0,...,T and {Bt }t=0,...,T that can be considered as the processes of the value of money in time, it is not clear how to define discounted price processes of risky assets. The same reason makes it unclear how to define the notion of a martingale measure. To avoid this difficulty we propose the following concept of a martingale pair.

Definition 3.1 Let us assume that we are given the market M = (S, P) deﬁned on the probability space (Ω, F,P ) with the ﬁltration {Ft}t=0,...,T . ∗ If there exist a predictable process {Bt}t=0,...,T and a probability measure P on (Ω, FT ) such that (i) P ∗ is equivalent to P , Discrete-time market models from the small investor point of view [29]

(ii) B0 = 1 and for all t ∈ {0,...,T − 1} we have

+ − Bt+1 Bt+1 Bt+1 + ≤ ≤ − , Bt Bt Bt

1 d St St ∗ (iii) the process {( ,..., )}t=0,...,T is P -martingale, Bt Bt ∗ then we say that the pair ({Bt}t=0,...,T ,P ) is a martingale pair for the market M = (S, P).

Let us assume that we are given a predictable process {Bt}t=0,...,T satisfying condition (ii) of Definition 3.1. Then, we consider the model Mf = (S,˜ P˜), where ˜ 1 d ˜ the process S of prices is defined as {(Bt,St ,...,St )}t=0,1,...,T , the set P consists 0 1 d d+1 of all predictable processes {(Θt , Θt ,..., Θt )}t=0,1,...,T with values in R . 0 1 d The values of the strategy Θ = {(Θt , Θt ,..., Θt )}t=0,1,...,T in the model Mf = (S,˜ P˜) are defined by

Θ 0 1 1 d d Vt− = Θt Bt + Θt St + ... + Θt St for t = 0, 1,...,T and Θ 0 1 1 d d Vt+ = Θt+1Bt + Θt+1St + ... + Θt+1St for t = 0, 1,...,T. Similarly to the proof of Lemma 2.9, one can prove the following

Lemma 3.2 ˜ ˜ If there is a trading strategy {Θt}t=0,...,T , deﬁned on the market Mf = (S, P), such Θ Θ Θ Θ Θ that V0− = 0, Vt− ≥ Vt+ for t ∈ {0,...,T }, and P (VT + ≥ 0) = 1, P (VT + > 0) > 0, then there is an arbitrage opportunity on the market Mf = (S,˜ P˜). Using the above lemma, we can prove the following

Lemma 3.3 If there is an arbitrage opportunity on the market M, then there is, also, an arbitrage opportunity on the market Mf.

Proof. Let us assume that {Θt}t=0,...,T , defined on the market M = (S, P) is an arbitrage opportunity. Let us define a strategy {Ψt}t=0,...,T , defined on the market Mf, as follows 0 + + − − + − Ψ0 = Θ0 B0 − Θ0 B0 = Θ0 − Θ0 and + + − − 0 Θt Bt−1 − Θt Bt−1 Ψt = (18) Bt−1 for t ∈ {1,...,T } and i i Ψt = Θt (19) Ψ Θ for t ∈ {0,...,T } and i ∈ {1, . . . , d}. Observe that V0− = V0− = 0. We will Ψ Ψ 1 1 d d prove that Vt− ≥ Vt+, for t ∈ {0,...,T }. Denote Xt− = Θt St + ... + Θt St and [30] Marek Karaś and Anna Serwatka

1 1 d d Xt+ = Θt+1St + ... + Θt+1St . Since {Θt}t=0,...,T is a self-ﬁnancing strategy, we + + − − + + − − have Θt Bt − Θt Bt + Xt− = Θt+1Bt − Θt+1Bt + Xt+ and so

+ + + − − − Xt− − Xt+ = (Θt+1 − Θt )Bt − (Θt+1 − Θt )Bt . (20)

Observe that, by (18)–(20) and the condition (ii) of Deﬁnition 3.1, we get

Ψ Ψ 0 0 Vt− − Vt+ = Ψt Bt + Xt− − Ψt+1Bt − Xt+ + + − − + + − − Θt Bt−1 − Θt Bt−1 Θt+1Bt − Θt+1Bt = Bt − Bt Bt−1 Bt + + + − − − + (Θt+1 − Θt )Bt − (Θt+1 − Θt )Bt + − +Bt−1 + −Bt−1 − = Θt Bt − Bt − Θt Bt − Bt Bt−1 Bt−1 ≥ 0.

Also, by (18)–(20) and the condition (ii) of Deﬁnition 3.1, we obtain

+ + − − Ψ Ψ ΘT BT −1 − ΘT BT −1 VT = VT − = BT + XT − BT −1

+ + BT − − BT = ΘT BT −1 − ΘT BT −1 + XT − BT −1 BT −1 + + − − Θ ≥ ΘT BT − ΘT BT + XT − = VT − Θ = VT .

By the above inequality, we have

Ψ Θ P (VT ≥ 0) ≥ P (VT ≥ 0) = 1 and Ψ Θ P (VT > 0) ≥ P (VT > 0) = 1 and so by Lemma 3.2, there is an arbitrage opportunity on the market Mf = (S,˜ P˜).

Example 3.4 Let us consider a one-period model M of a financial market with one risky as- − t + t set St and two processes Bt = (1, 1) and Bt = (1, 02) . The stock price at time t = 1 can take two different values S1(ω1) = 108, S1(ω2) = 104 and ˜ ˜ ˜ S0 = 100. Next, we consider the model Mf = (S, P), where the process S of prices is defined as {(Bt,St)}t=0,1,...,T , the set P˜ consists of all predictable processes 0 t {(Θt , Θt)}t=0,1,...,T and Bt = (1, 02) .

0 1 Let us consider a strategy Θ on the market Mf deﬁned (Θ0, Θ0) = (0, 0) and 0 1 0 (Θ1, Θ1) = (−100, 1). Note that Θ is a self-ﬁnancing strategy, because Θ0B0 + 1 0 1 Θ Θ Θ0S0 = Θ1B0 + Θ1S0 and an arbitrage strategy, because V0 = 0 and V1 = 0 1 Θ1B1 + Θ1S1 = −100 + S1 > 0. Discrete-time market models from the small investor point of view [31]

We will show that there is not an arbitrage strategy on the market M. Suppose + − that ϕ is an arbitrage strategy on the market M, then ϕ satisﬁes ϕ1 ≥ 0, ϕ1 ≥ 0,

+ − 1 + − 1 ϕ0 − ϕ0 + ϕ0S0 = 0, ϕ1 − ϕ1 + ϕ1S0 = 0 (21) and + − 1 1, 02ϕ1 − 1, 1ϕ1 + ϕ1S1 > 0. (22)

− 1 − 1 Using (21)–(22) we have (ϕ1 −100ϕ1)1, 02−1, 1ϕ1 +ϕ1S1 > 0, which is equivalent 1 − (S1−100)ϕ1 1 − to ϕ1 < 0,08 . If ϕ1 ≤ 0 we obtain a contradiction with ϕ1 ≥ 0, so we need 1 1 1 + − 1 (S1−100)ϕ1 1 (S1−108)ϕ1 ϕ1 > 0. Then ϕ1 = ϕ1 − 100ϕ1 < 0,08 − 100ϕ1 = 0,08 ≤ 0, which + + − contradicts ϕ1 ≥ 0. We have proved that there is no (ϕt , ϕt ) satisfying conditions + − (21)–(22) and ϕ1 ≥ 0, ϕ1 ≥ 0, so the market M is arbitrage free. Now, we are in a position to prove the ﬁrst main result.

Theorem 3.5 If there exists a martingale pair for the market M = (S, P), then there is no arbitrage opportunity on the market M = (S, P).

∗ Proof. Let us assume that there exists a martingale pair ({Bt}t=0,...,T ,P ) for the 1 d St St ∗ market M = (S, P). Then the process {( ,..., )}t=0,...,T is P -martingale. Bt Bt This means that P ∗ is a martingale measure in the model Mf. Using The First Fundamental Theorem of Asset Pricing we show that Mf is arbitrage-free. It follows from Lemma 3.3 that there is not any arbitrage opportunity on the market M.

We also have the following result.

Theorem 3.6 ∗ If ({Bt}t=0,...,T ,P ) is a martingale pair for the market M = (S, P), then for any 2 h ∈ L (Ω, FT ,P ) letting −1 Ct = EP ∗ (BT Bth | Ft) the extended model M¯ = (S,¯ P¯) is arbitrage free, where

¯ + − 1 d S = {(Bt ,Bt ,St ,...,St ,Ct)}t=0,1,...,T

¯ + − 1 d d+1 and P is the set of all predictable processes {(Θt , Θt , Θt ,..., Θt , Θt )}t=0,1,...,T with values in [0, +∞)2 × Rd+1.

1 d St St ∗ Ct Proof. Observe that the process {( ,..., )}t=0,...,T is P -martingale and = Bt Bt Bt 1 d −1 ∗ St St Ct ∗ EP ∗ (B h | Ft) is also P -martingale, so {( ,..., , )}t=0,...,T is P -martin- T Bt Bt Bt ∗ gale. Hence ({Bt}t=0,...,T ,P ) is a martingale pair for the market M¯ = (S,¯ P¯). Thus, we can use Theorem 3.5. [32] Marek Karaś and Anna Serwatka

4. Some special cases

In this section we will examine some special cases in which the implication of Theorem 3.5 can be replaced by equivalence. We will start with the easiest; a one-period two-state model of a ﬁnancial market with one risky asset St and two diﬀerent deterministic interest rates for loans (rl) and deposits (rd).

Example 4.1 Let us assume that the probability space (Ω, F,P ) is given as follows: Ω = {u, d}, F = P(Ω) and P (u),P (d) > 0 with P (u) + P (d) = 1. We also consider ﬁltration {F}t=0,1 with F0 = {∅, Ω}, F1 = F, and assume that the process + − + − + {(Bt ,Bt ,St)}t=0,1 is given by B0 = B0 ≡ 1, S0 ∈ (0, +∞), B1 ≡ 1 + rd, − B1 ≡ 1 + rl, where positive numbers rd and rl satisfy the obvious relation rd < rl, u d d u and S1(u) = S1 , S1(d) = S1 with 0 < S1 < S1 . Of course rd and rl denote, respectively, the interest rates under which the bank account and the bank loans are subjected. u One can easily check that there is an arbitrage opportunity if S0(1 + rd) ≥ S1 d or S1 ≥ S0(1 + rl) (see Deﬁnition 2.8) In other words, the necessary conditions for the considered market to be arbitrage free are

u d S0(1 + rd) < S1 and S1 < S0(1 + rl).

The following easy lemma will be used to prove the existence of a martingale pair for the arbitrage free models considered in this section.

Lemma 4.2 Let us consider four positive numbers d, u, rd, rl such that d < u, rd < rl, 1 + rd < u and d < 1 + rl. Then, we have the following inequality

max{d, 1 + rd} < min{u, 1 + rl}.

Proof. There are two cases: d ≥ 1 + rd or d < 1 + rd. In the ﬁrst case, we have max{d, 1 + rd} = d. Since, by assumptions, d < u and d < 1 + rl, it follows that max{d, 1 + rd} = d < min{u, 1 + rl}. In the second case, it is true that max{d, 1 + rd} = 1 + rd and so max{d, 1 + rd} = 1 + rd < min{u, 1 + rl}, because per the assumptions we have 1 + rd < 1 + rl and 1 + rd < u.

Now we will provide the necessary and suﬃcient conditions for the market of Example 4.1 to be arbitrage free. In other words, in the context of Example 4.1, we reverse the implication of Theorem 3.5.

Theorem 4.3 Let us consider the model of the ﬁnancial market described in Example 4.1. Then, the following conditions are equivalent (a) the model is arbitrage free, u d (b) S0(1 + rd) < S1 and S1 < S0(1 + rl), (c) the model permits a martingale pair. Discrete-time market models from the small investor point of view [33]

Proof. Since the implication (a)⇒(b) was already mentioned above in Example 4.1 (the necessary conditions) and the implication (c)⇒(a) is a consequence of Theo- rem 3.5, thus we only need to show the implication (b)⇒(c). Now, assume that the condition (b) is satisﬁed. Then, by Lemma 4.2 for d = S1(d) and u = S1(u) , there exists a positive number r such that S0 S0

nS1(d) o nS1(u) o max , 1 + rd < 1 + r < min , 1 + rl . (23) S0 S0

Let us consider the process {Bt}t=0,1 given by B0 ≡ 1, B1 ≡ 1+r and the function Q: F → R given by S (1 + r) − S (d) S (u) − S (1 + r) Q(u) = 0 1 and Q(d) = 1 0 . S1(u) − S1(d) S1(u) − S1(d)

Since, by (23), S1(d) < S0(1 + r) < S1(u), one can see that Q(u),Q(d) > 0. We also see, by the deﬁnition of Q, that Q(u) + Q(d) = 1. Thus, the function Q is a probability measure, and moreover this probability measure is equivalent to P (because we have Q(u),Q(d),P (u),P (d) > 0). From (23), we also obtain

+ − B1 B1 B1 + = 1 + rd < 1 + r = = 1 + r < 1 + rl = − . B0 B0 B0

S1 S0 One can, also, easily see that EQ( ) = , where EQ denotes the mean value B1 B0 St with respect to probability measure Q. But, this means that the process { }t=0,1 Bt is a Q-martingale. So we are checked that the pair ({Bt}t=0,1,Q) is a martingale pair for the considered market and thus the proof of implication (b)⇒(c) is ﬁnished.

The next special case that we will examine is the following one-period multi- state model.

Example 4.4 Now, we assume that the probability space (Ω, F,P ) is given as follows: Ω = {ω1, . . . , ωn}, F = P(Ω) and P (ω1),...,P (ωn) > 0 with P (ω1) + ... + P (ωn) = 1 for some n ≥ 2. We also consider ﬁltration {F}t=0,1 with F0 = {∅, Ω}, F1 = F. + − + − Assume that the process {(Bt ,Bt ,St)}t=0,1 is given by B0 = B0 ≡ 1, S0 ∈ + − (0, +∞), B1 ≡ 1 + rd, B1 ≡ 1 + rl, where positive numbers rd and rl satisfy rd < rl and S1(ω1),...,S1(ωn) are positive numbers. Without loss of generality, we can assume that S1(ω1) < S1(ω2) < . . . < S1(ωn). One can easily check that there is an arbitrage opportunity if S0(1 + rd) ≥ S1(ωn) or S1(ω1) ≥ S0(1 + rl). In other words, the necessary conditions for the considered market to be arbitrage free are

S0(1 + rd) < S1(ωn) and S1(ω1) < S0(1 + rl).

Now, we prove the following generalization of Theorem 4.3. [34] Marek Karaś and Anna Serwatka

Theorem 4.5 Let us consider the model of the ﬁnancial market described in Example 4.4. Then, the following conditions are equivalent (a) the model is arbitrage free,

(b) S0(1 + rd) < S1(ωn) and S1(ω1) < S0(1 + rl), (c) the model permits a martingale pair.

Proof. As in the proof of Theorem 4.3, we only need to show the implication (b)⇒(c). Thus, assume that the condition (b) is satisﬁed. By Lemma 4.2 for d = S1(ω1) and u = S1(ωn) , there exists a positive number r such that S0 S0

nS1(ω1) o nS1(ωn) o max , 1 + rd < 1 + r < min , 1 + rl . S0 S0

Moreover, we can choose r such that 1 + r∈ / { S1(ω1) ,..., S1(ωn) }. Let k ∈ S0 S0 {1, . . . , n − 1} be such that S (ω ) S (ω ) 1 k < 1 + r < 1 k+1 . S0 S0 Without loss of generality, we can assume that k ≤ n − k (if the inequality k > n−k holds, the argument is similar). Now, we can choose a partition I1,...,Ik of the set {k + 1, . . . , n} with Il 6= ∅ for l = 1, . . . , k. Now, let us consider the process {Bt}t=0,1 given by B0 ≡ 1 and B1 ≡ 1 + r and the function Q: F → R given by P |S1(ωi) − S0(1 + r)| Q(ω ) = i∈Il l Pk Pn i=1 #Ii|S1(ωi) − S0(1 + r)| + i=k+1 |S1(ωi) − S0(1 + r)| for l = 1, . . . , k and |S (ω ) − S (1 + r)| Q(ω ) = 1 j 0 l Pk Pn i=1 #Ii|S1(ωi) − S0(1 + r)| + i=k+1 |S1(ωi) − S0(1 + r)| for l ∈ Ij. From the deﬁnition of Q it is obvious that Q(ω1),...,Q(ωn) > 0. One can, also, check that Q(ω1)+...+Q(ωn) = 1. Thus, the function Q is a probability measure that is equivalent to P . The same reasons, as in the proof of Theorem 4.3 + − B1 B1 B1 S1 S0 give + < < − . To see that EQ( ) = S0 = , we make the following B0 B1 B0 B0 B0 calculations n n S1 X S1(ωi) X S0(1 + r) + [S1(ωi) − S0(1 + r)] E = Q(ω ) = Q(ω ) Q B 1 + r i 1 + r i 1 i=1 i=1 k 1 X = S + [S (ω ) − S (1 + r)]Q(ω ) 0 1 + r 1 i 0 i i=1 X + [S1(ωj) − S0(1 + r)]Q(ωj)

j∈Ii Discrete-time market models from the small investor point of view [35] and notice that, for all i ∈ {1, . . . , k}, we have X [S1(ωi) − S0(1 + r)]Q(ωi) + [S1(ωj) − S0(1 + r)]Q(ωj) = 0.

j∈Ii

So we have checked that the pair ({Bt}t=0,1,Q) is a martingale pair for the considered market and have also ﬁnished the proof of implication (b)⇒(c). Before we present, in the next section, a general result for ﬁnite models, we will examine the next model which is similar to the model of Cox-Ross-Rubinstein. We will call it a CRR-type model.

Example 4.6 This model can be realized on the probability space (Ω, F,P ) defined as follows: T Ω = {ω1, . . . , ωn} with n = 2 , where T is the time horizon, F = P(Ω) and P (ω1),...,P (ωn) > 0 with P (ω1) + ... + P (ωn) = 1. 1 1 We also consider filtration {F}t=0,...,T with F0 = {∅, Ω}, F1 = σ({Au,Ad}), 1 1 2 2 2 2 where A = {ω , . . . , ω n }, A = {ω n , . . . , ω }, F = σ({A ,A ,A ,A }), d 1 2 u 2 +1 n 2 uu ud du dd 2 2 2 where A = {ω , . . . , ω n }, A = {ω n , . . . , ω n }, A = {ω n , . . . , ω n } and dd 1 4 du 4 +1 2 ud 2 +1 3· 4 2 A = {ω n , . . . , ω }, and so on. To be precise, for k ∈ {1,...,T }, we define uu 3· 4 +1 n F as the σ-field generated by the partition of the set Ω in to the 2k subsets Ak k ε1...εk k n n with ε1, . . . , εk ∈ {u, d}, where Aε ...ε = {ωϕk(ε1...εk)· +1, . . . , ω(ϕk(ε1...εk)+1)· } 1 k 2k 2k and ϕk(ε1 . . . εk) = ε1 . . . εk(2), assuming the value of the k-digit binary sequence ε1 . . . εk, where we assign value 1 to the ’digit’ u and value 0 to the ’digit’ d. + − To define the process {(Bt ,Bt ,St)}t=0,...,T , we assume that u and d denote, depending on the context, symbols acting as short-cuts of up and down (like in the above-mentioned definition of filtration), and positive numbers (like below in the definition of the process {St}t=0,...,T ). Of course, if u and d are considered as positive numbers, we assume that d < u. + − + t Now, we can define {(Bt ,Bt ,St)}t=0,...,T . First of all, we put Bt ≡ (1 + rd) − t and Bt ≡ (1 + rl) for t = 0, 1,...,T and X Sk = ε1 ··· εkS0 · 1Ak , ε1...εk ε,...,εk∈{u,d}

k where 1Ak denotes the characteristic function of the set Aε ...ε . One can ε1...εk 1 k easily verify that there is an arbitrage opportunity if 1 + rd ≥ u or d ≥ 1 + rl. In other words, the necessary conditions for the considered market to be arbitrage free are 1 + rd < u and d < 1 + rl. The requirements for the model in Example 4.6 to be arbitrage free are the following.

Theorem 4.7 Let us consider the model of the ﬁnancial market described in Example 4.6. Then, the following conditions are equivalent (a) the model is arbitrage free, [36] Marek Karaś and Anna Serwatka

(b)1+ rd < u and d < 1 + rl, (c) the model permits a martingale pair.

Proof. As in the proofs of Theorems 4.3 and 4.5, we only need to show the implication (b)⇒(c). So choose a positive number r such that

max{d, 1 + rd} < 1 + r < min{u, 1 + rl}.

Using this number r, we deﬁne (1 + r) − d u − (1 + r) p = and q = 1 − p = . u − d u − d

By the definition of the filtration {Ft}t=0,...,T , we have for any k ∈ {1,...,T − 1} and for any ε , . . . , ε ∈ {u, d} that Ak Ak+1 ∪ Ak+1 . We also have 1 k ε1...εk ε1...εkd ε1...εku 1 1 Ω = Ad ∪ Au. Thus, the probability measure Q: F → R, we can define such that 1 1 Q(Au) = p, Q(Ad) = q and for any k ∈ {1,...,T − 1} and any ε1, . . . , εk ∈ {u, d}, QAk+1 | Ak = p and QAk+1 | Ak = q. ε1...εku ε1...εk ε1...εkd ε1...εk By this definition of Q, as one can check, we have for any k ∈ {1,...,T − 1} and any ε1, . . . , εk ∈ {u, d}, such that

Sk+1 k ε1 ··· εkuS0 ε1 ··· εkdS0 ε1 ··· εkS0 Sk EQ Aε ...ε = · p + · q = , B 1 k (1 + r)k+1 (1 + r)k+1 (1 + r)k B Ak k+1 k ε1...εk

Sk+1 Sk which means that EQ( | Fk) = for k = 1,...,T − 1. Of course, we also Bk+1 Bk S1 S1 S0 have EQ( | F0) = EQ( ) = . B1 B1 B0 + − Bt+1 Bt+1 Bt+1 Since, also, + = 1 + rd < 1 + r = B = 1 + r < 1 + rl = − for any Bt t Bt t ∈ {0,...,T − 1}, it follows that the pair ({Ft}t=0,...,T ,Q) is a martingale pair for the model.

5. The first fundamental-type theorem for finite models with two different interest rates

Example 5.1 Let us consider a model of a financial market with one risky asset St. Assume + − that {Bt }t=0,1,...,T and {Bt }t=0,1,...,T are predictable stochastic processes, with + values of (0, +∞), such that the process {Bt }t=0,1,...,T is a deposit process and − the process {Bt }t=0,1,...,T is a loan process (see Definition 2.1). This model can be realized on the finite probability space (Ω, F,P ) defined as follows: Ω = {ω1, . . . , ωn}, T ∈ N+ is the time horizon, F = P(Ω) and P (ω1),...,P (ωn) > 0 with P (ω1) + ... + P (ωn) = 1. We also define filtration {Ft}t=0,...,T on the measurable space (Ω, F) as follows F0 = {∅, Ω} and the filtration {Ft}t=0,...,T is described by the sequence of partitions (t) (t) A = {Ai | i = 1, . . . , rt} for all t ∈ {0,...,T } Discrete-time market models from the small investor point of view [37]

(0) with A = {Ω} (and r0 = 1), such that for all t ∈ {0,...,T − 1} there is (t) (t) a partition {I1 ,...,Irt } of the set {1, . . . , rt+1} such that for all i ∈ {1, . . . , rt}, (t) S (t+1) we have Ai = (t) Aj . We also assume that FT = F. j∈Ii Each set of the partition A(t) represents one of the possible states of the world at time t, and the number rt can be interpreted as the number of states in which the world can arrive at the moment t. Let us ﬁx t ∈ {0,...,T − 1} and + − i ∈ {1, . . . , rt}. Since the functions Bt ,Bt ,St are constant on the sets of the (t) + (t) form Ai , by Bt (Ai ) and so on we will denote this constant value. Note that + (t+1) + (t) − (t+1) − (t) (t) Bt+1(Aj ) = Bt+1(Ai ) and Bt+1(Aj ) = Bt+1(Ai ) for all j ∈ Ii . One can check that the necessary conditions for the considered market to be arbitrage free are

(t+1) + (t) B (A ) maxj∈I(t) {St+1(Aj )} t+1 i < i + (t) (t) Bt (Ai ) St(Ai ) and (t+1) − (t) B (A ) minj∈I(t) {St+1(Aj )} t+1 i > i , − (t) (t) Bt (Ai ) St(Ai ) for all t ∈ {0,...,T − 1} and all i ∈ {1, . . . , rt}. The necessary and suﬃcient conditions for the model of Example 5.1 to be arbitrage free are the following.

Theorem 5.2 Let us consider the model of ﬁnancial market described in Example 5.1. Then, the following conditions are equivalent: (a) the model is arbitrage free , (t+1) (t+1) + (t) max (t) {St+1(A )} − (t) min (t) {St+1(A )} B (A ) j∈I j B (A ) j∈I j t+1 i i t+1 i i (b) + (t) < (t) and − (t) > (t) for Bt (Ai ) St(Ai ) Bt (Ai ) St(Ai ) all t ∈ {0,...,T − 1} and all i ∈ {1, . . . , rt}. (c) the model permits a martingale pair.

(1) Proof. We only need to show the implication (b)⇒(c). For t = 1 we choose r1 (1) and k1 as in Theorem 4.5, such that

(1) + (1) − nmins S1(As ) B1 o (1) nmaxs S1(As ) B1 o max , + < 1 + r1 < min , − . S0 B0 S0 B0

Consider the process {Bt}t=0,1, and the function Q on F1 deﬁned analogously as in Theorem 4.5. Let us ﬁx t ∈ {2,...,T − 1} and i ∈ {1, . . . , rt}. By analogy we (t) choose ri such that

(t) min (t) S (A ) + (t−1) j∈I t j B (A ) max i , t i < 1 + r(t) (t−1) + (t−1) i St−1(Ai ) Bt−1(Ai ) [38] Marek Karaś and Anna Serwatka

(t) max (t) S (A ) − (t−1) j∈I t j B (A ) < min i , t i . (t−1) − (t−1) St−1(Ai ) Bt−1(Ai )

(s−1) (s−1) Let the process {Bs}s=0,...,t be given by B0 ≡ 1 and Bs(Aj ) = Bs−1(Aj )(1+ (s) (s) rj ) for s ∈ {1, . . . , t} and j ∈ Ii . (t+1) (t) Now, we consider the conditional probability Q(Aj | Ai ) for all t ∈ {1,..., (i) T − 1}, all i ∈ {1, . . . rt} and all j ∈ It , and define it analogously as in the proof of Theorem 4.5. The definition of Q on F1 gives the definition of Q: FT → R. By this definition of Q (similar to the proof in Theorem 4.5) we get for any t = 1,...,T − 1 and for any i ∈ {1, . . . , rt},

(t+1) St+1 X St+1(Aj ) (t+1) (t) St E At = · Q(A | A ) = , Q i (t+1) j i (t) Bt+1 Bt A (t) Bt+1(Aj ) i j∈Ii

St St−1 which means that EQ( | Ft−1) = for t = 2,...,T . Of course, we also have Bt Bt−1 S1 S1 S0 EQ( | F0) = EQ( ) = . B1 B1 B0 + − Bt+1 Bt+1 Bt+1 Since, also, + < B < − for any t ∈ {0,...,T − 1}, it follows that the Bt t Bt pair ({Ft}t=0,...,T ,Q) is a martingale pair for the model. We will show the application of Theorem 5.2, considering the phenomenon of diﬀerent access to arbitration in a certain sense in the same market for two diﬀerent investors.

Example 5.3 Let us assume that the probability space (Ω, F,P ) is given as follows Ω = {u, d}, F = P(Ω) and P (u),P (d) > 0 with P (u) + P (d) = 1. Let {F}t=0,1 be filtration + − such that F0 = {∅, Ω}, F1 = F, and the process {(Bt ,Bt ,Bt,St)}t=0,1 be given + − + − by B0 = B0 = B0 ≡ 1, S0 ∈ (0, +∞), B1 ≡ 1 + rd, B1 ≡ 1 + rl and B1 = + B1 , where positive numbers rd and rl, because of their interpretation, satisfy the obvious relation rd < rl. Finally let S1(u) > S1(d) > 0 be real numbers. Of course rd and rl denote, respectively, the interest rates under which the bank account and the bank loans are subjected. Furthermore, we assume that S1(d) S1(u) 1 + rd < < 1 + rl < . We also make an assumption that a small player S0 S0 can not take any position in Bt while a big player can take any position in Bt (including a short position). From the small player’s point of view the considered model is indifferent from + − the model {(Bt ,Bt ,St)}t=0,1 without any constraints. This model by Theo- rem 4.3, is arbitrage free and so the small player does not have an arbitrage + − opportunity in the model {(Bt ,Bt ,Bt,St)}t=0,1. From the big player’s point of view the situation is completely different because of the possibility of taking a position in Bt, especially a short position, means that the big player can borrow money at the same interest rate as he can make a deposit. S1(d) S1(u) Since 1 + rd < < , the big player has an arbitrage opportunity in the S0 S0 model. Discrete-time market models from the small investor point of view [39]

The Example 5.3 shows that a big player who has the same interest rate for deposits and loans is in a prime position compared to a small player who has two different interest rates. Moreover, we can create a market model in which both players have two different interest rates, but the big player is still in a better position. We consider that situation in the next example. Example 5.4 Let us assume that the probability space (Ω, F,P ) and the filtration {Ft}t=0,1 is given in the previous example. Now, consider the following process

+ − + − {(Bt,s,Bt,s,Bt,b,Bt,b,St)}t=0,1 (24)

+ − + − + − + with B0,s = B0,s = B0,b = B0,b ≡ 1, B1,s ≡ 1+rd,s,B1,s ≡ 1+rl,s, B1,b ≡ 1+rd,b, − B1,b ≡ 1 + rl,b, where positive numbers rd,s, rl,s, rd,b, rl,b satisfy the following relation rd,s ≤ rd,b ≤ rl,b ≤ rl,s, S0 ∈ (0, +∞) and let S1(u) > S1(d) > 0 be real + numbers. We also assume that the small player cannot take any position in Bt,b − + − and Bt,b. While the big player can take long position not only in Bt,s and Bt,s (as + − the small player can) but also in Bt,b and Bt,b (both kinds of players can take long and short position in St). From the small player’s point of view the considered + − model gives exactly the same possibilities as the model {(Bt,s,Bt,s,St)}t=0,1 gives. While from the big player’s point of view the considered model gives even more + − possibilities from the model {(Bt,b,Bt,b,St)}t=0,1. Now, suppose that one of the following is satisﬁed

S1(d) S1(u) 1. 1 + rd,s ≤ 1 + rd,b ≤ 1 + rl,b ≤ < 1 + rl,s < , S0 S0 S1(d) S1(u) 2. < 1 + rd,s < ≤ 1 + rd,b ≤ 1 + rl,b < 1 + rl,s. S0 S0 + − Now, using Theorem 4.3 for the model {(Bt,s,Bt,s,St)}t=0,1 we obtain, that the small player does not have an arbitrage opportunity in the model (24). By The- + − orem 4.3 using the model {(Bt,b,Bt,b,St)}t=0,1 and the above discussion, we conclude that a big player does have an arbitrage opportunity in the model. Remark 5.5 Note that we can generalize Example 5.4 and consider the market model, as in Example 5.1, with two players with different deposit processes and loan processes. With properly selected processes the small player doesn’t have an arbitrage opportunity, because of Theorem 5.2, in contrast to the big player, who has an arbitrage opportunity. The examples show that different investors, who have different access to deposits and loans, have different positions related to arbitrage.

References

[1] Dalang, Robert C., and Andrew Morton, and Walter Willinger. "Equivalent martingale measures and no-arbitrage in stochastic securities market models." Stochas- tics Stochastics Rep. 29, no. 2 (1990): 185–201. Cited on 17 and 18. [2] Delbaen, Freddy, and Walter Schachermayer. "A general version of the fundamental theorem of asset pricing." Math. Ann. 300, no. 3 (1994): 463–520. Cited on 18. [40] Marek Karaś and Anna Serwatka

[3] Delbaen, Freddy, and Walter Schachermayer. "The fundamental theorem of asset pricing for unbounded stochastic processes." Math. Ann. 312, no. 2 (1998): 215– 250. Cited on 18. [4] Harrison, J. Michael, and David M. Kreps. "Martingales and arbitrage in multi- period securities markets." J. Econom. Theory 20, no. 3 (1979): 381–408. Cited on 18. [5] Harrison, J. Michael, and Stanley R. Pliska. "Martingales and stochastic integrals in the theory of continuous trading." Stochastic Process. Appl. 11, no. 3 (1981): 215–260. Cited on 18. [6] Jacod, Jean, and Al’bert N. Shiryaev. "Local martingales and the fundamental asset pricing theorems in the discrete-time case." Finance Stoch. 2, no. 3 (1998): 259–273. Cited on 18. [7] Jensen, Bjarne A. "Valuation before and after tax in the discrete time, finite state no arbitrage model." Ann. Finance 5, no. 1 (2009): 91–123. Cited on 18. [8] Jouini, Elyes, and Hédi Kallal. "Martingales and arbitrage in securities markets with transaction costs." J. Econom. Theory 66, no. 1 (1995): 178–197. Cited on 18. [9] Kabanov, Yuri, and Miklós Rásonyi, and Christophe Stricker. "No-arbitrage cri- teria for financial markets with efficient friction." Finance Stoch. 6, no. 3 (2002): 371–382. Cited on 18. [10] Kabanov, Yuri, and Mher Safarian. Markets with transaction costs. Mathematical theory. Springer Finance. Berlin: Springer-Verlag, 2009. Cited on 18. [11] Rola, Przemysław. "Arbitrage in markets with bid-ask spreads: the fundamental theorem of asset pricing in finite discrete time markets with bid-ask spreads and a money account." Ann. Finance 11, no. 3-4, (2015): 453–475. Cited on 18. [12] Schachermayer, Walter. "The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time." Math. Finance 14, no. 1 (2004): 19–48. Cited on 18.

Marek Karaś AGH University of Science and Technology Faculty of Applied Mathematics al. A. Mickiewicza 30 30-059 Kraków Poland E-mail: [email protected]

Anna Serwatka Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University in Kraków ul. Łojasiewicza 6 30-348 Kraków Poland E-mail: [email protected]

Received: March 14, 2016; ﬁnal version: May 12, 2017; available online: July 4, 2017. Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 41-50 DOI: 10.1515/aupcsm-2017-0003

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Ioannis K. Argyros and Santhosh George Local convergence of a multi-step high order method with divided diﬀerences under hypotheses on the ﬁrst derivative Communicated by Tomasz Szemberg

Abstract. This paper is devoted to the study of a multi-step method with divided diﬀerences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the ﬁrst derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

1. Introduction

Grau et. al. in [12], studied a sixth-order multi-step method deﬁned for each n = 0, 1, 2,... by

−1 yn = xn − An F (xn), −1 zn = yn − Bn F (yn), (1) −1 xn+1 = zn − Bn F (zn), where An = [un, vn; F ], Bn = 2[yn, xn; F ] − [un, vn; F ], un = xn + F (xn) and ∗ vn = xn − F (xn), for approximating a solution x of the equation F (x) = 0, (2)

AMS (2010) Subject Classification: 65D10, 49M15, 74G20, 41A25. Keywords and phrases: Multi-step method, restricted convergence domain, radius of convergence, local convergence. [42] Ioannis K. Argyros and Santhosh George where F : D ⊆ B1 → B2 is a Fréchet differentiable operator between Banach 2 spaces B1, B2 and [., .; F ] is a divided difference of order one on D . Due to the wide applications, finding a solution for (2) is an important problem in applied mathematics. Most of the solution methods for solving (2) are iterative and for iterative methods order of convergence is an important issue. Convergence analysis of higher order iterative methods require assumptions on the higher order Fréchet derivatives of the operator F . That restricts the applicability of these methods. i Notice that in [12] B1 = B2 = R (i a natural integer). However, we study method (2) in the more general setting of a Banach space. We also provide com- ∗ putable radius of convergence and error bounds on kxn − x k based on Lipschitz constants not given in [12]. The study of the local convergence in this way is also important because it provides the difficulty in choosing the initial points. Other- wise as in the earlier studies the choice of the initial point is a shot in the dark. Throughout this paper L(B2, B1) denotes the set of bounded linear operators between B1 and B2 and B(z, ρ), B¯(z, ρ) stand, respectively for the open and closed balls in B1 with center z ∈ B1 and of radius ρ > 0. Convergence analysis in [12] is based on the assumptions on the Fréchet derivative F up to the order six. In this study we use only assumptions on the first Fréchet derivative of the operator F in our convergence analysis, so that the method (1) can be applied to solve equations but the earlier results cannot be applied [1, 2, 3, 4, 5, 18, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 9, 19, 20, 21, 22] (see Example 3.2). The rest of the paper is structured as follows. In Section 2 we present the local convergence analysis of the method (1). We also provide a radius of convergence, computable error bounds and a uniqueness result. Numerical examples are given in the last section.

2. Local convergence

The local convergence analysis of (1) is based on some parameters and scalar 2 functions. Let α ≥ 0, β ≥ 0 be parameters and ω0 : [0, +∞) → [0, +∞) be a continuous nondecreasing function satisfying ω0(0, 0) = 0. Deﬁne parameter r0 by

r0 = sup{t ≥ 0 : ω0(αt, βt) < 1}. (3)

2 Let ϕ0 : [0, r0) → [0, +∞), ω, ω1 : [0, r0) → [0, +∞) be continuous and nondecreasing functions. Deﬁne functions g1 and h1 on the interval [0, r0) by

ω1(ϕ0(t)t, ϕ0(t)t) g1(t) = 1 − ω0(αt, βt) and

h1(t) = g1(t) − 1. Suppose that

ω1(0, 0) < 1. (4) Local convergence of a multi-step high order method [43]

We have by (4) that

ω1(0, 0) h1(0) = − 1 < 0, (ω0(0, 0) = 0) 1 − ω0(0, 0)

− and by (3) h1(t) → +∞ as t → r0 . The intermediate value theorem assures the existence of a solution for equation h1(t) = 0 in (0, r0). Denote by r1 the smallest such solution. Deﬁne also functions p and hp on [0, r0) by

p(t) = ω0(g1(t)t, t) + ω((g1(t) + α)t, ϕ0(t)t) and hp(t) = p(t) − 1. Suppose that ω(0, 0) < 1. (5) We get hp(0) = ω0(0, 0) + ω(0, 0) − 1 < 0 − and hp(t) → +∞ as t → r0 . Denote by rp the smallest solution of equation hp(t) = 0. Let ϕ: [0, rp) → [0, +∞) be a continuous and nondecreasing function. Deﬁne functions g2 and h2 on the interval [0, rp) by

ϕ(g (t)t) g (t) = 1 + 1 g (t) 2 1 − p(t) 1 and h2(t) = g2(t) − 1. Suppose that ϕ(0) 1 + ω (0, 0) < 1. 1 − p(0) 1 − We obtain that h2(0) < 0 and h2(t) → +∞ as t → rp . Denote by r2 the smallest 2 solution of equation h2(t) = 0 on the interval (0, rp). Let ω : [0, r0) → [0, +∞) be a continuous and nondecreasing function. Deﬁne functions g3 and h3 on the interval [0, rp) by ϕ(g (t)t) g (t) = 1 + 2 g (t) 3 1 − p(t) 2 and h3(t) = g3(t) − 1. Suppose that ϕ(0) 1 + 1 + ϕ(0) ω (0, 0) < 1. (6) 1 − p(0) 1 − We get by (6) that h3(0) < 0 and h3(t) → +∞ as t → r0 . Denote by r3 the smallest solution of equation h3(t) = 0. Deﬁne the radius of convergence r by

r = min{ri, i = 1, 2, 3}. (7) [44] Ioannis K. Argyros and Santhosh George

Then, for each t ∈ [0, r) we have

0 ≤ gi(t) < 1 (8) and 0 ≤ p(t) < 1 (9) Define parameter R∗ by R∗ = max{αr, βr, r}. (10) The local convergence analysis of method (1) follows under the previous notation. Theorem 2.1 Let F :Ω ⊂ B1 → B2 be a continuously Fréchet differentiable operator and let 2 2 [., .; F ]:Ω → L(B1, B2) be a divided difference of order one on Ω for F . Sup- ∗ 2 pose there exists x ∈ Ω and function ω0 : [0, +∞) → [0, +∞) continuous and nondecreasing with ω0(0, 0) = 0, such that for each x, y ∈ Ω, F (x∗) = 0,F 0(x∗) is invertible (11) and 0 ∗ −1 0 ∗ ∗ ∗ kF (x ) ([x, y; F ] − F (x ))k ≤ ω0(kx − x k, ky − x k). (12) ∗ 2 Let Ω0 = Ω ∩ B(x , r0). There exist α ≥ 0, β ≥ 0, functions ω, ω1 : [0, r0) → [0, +∞), ϕ0, ϕ: [0, r0) → [0, +∞) continuous and nondecreasing such that for each x, y, u ∈ Ω0, 0 ∗ −1 ∗ ∗ kF (x ) ([x, y; F ] − [u, x ; F ])k ≤ ω1(kx − uk, ky − x k), (13) kF 0(x∗)−1([x, y; F ] − [u, z; F ])k ≤ ω(kx − uk, ky − zk), (14) ∗ ∗ k[x, x ; F ]k ≤ ϕ0(kx − x k), (15) kF 0(x∗)−1[x, x∗; F ]k ≤ ϕ(kx − x∗k), (16) B¯(x∗,R∗) ⊆ Ω, (17) kI + [x, x∗; F ]k ≤ α, kI − [x, x∗; F ]k ≤ β

∗ and conditions (5), (6) hold, where r0, r, R are defined by (3), (7) and (10), ∗ ∗ respectively. Then, the sequence {xn} generated for x0 ∈ U(x , r)−{x } by method (1) is well defined, remains in B(x∗, r) for each n = 0, 1, 2,... and converges to x∗. Moreover, the following estimates hold ∗ ∗ ∗ ∗ kyn − x k ≤ g1(kxn − x k)kxn − x k ≤ kxn − x k < r, (18) ∗ ∗ ∗ ∗ kzn − x k ≤ g2(kxn − x k)kxn − x k ≤ kxn − x k, (19) ∗ ∗ ∗ ∗ kxn+1 − x k ≤ g3(kxn − x k)kxn − x k ≤ kxn − x k, (20) where the functions gi for i = 1, 2, 3 are defined previously. Furthermore, if there exists R1 ≥ r such that

ω0(R1, 0) < 1 or ω0(0,R1) < 1, ∗ then the limit point x is the only solution of equation F (x) = 0 in Ω1 := Ω ∩ ∗ B(x ,R1). Local convergence of a multi-step high order method [45]

Proof. The estimates (18)–(20) shall be shown using induction. First we show that A0 is invertible, so y0 is then well deﬁned by the ﬁrst substep of method (1) for n = 0. Using (3), (11) and (12), we have that

0 ∗ −1 0 ∗ kF (x ) (A0 − F (x ))k ∗ ∗ ≤ ω0(kx0 − x − F (x0)k, kx0 − x + F (x0)k) ∗ ∗ ∗ ∗ ≤ ω0(k(I + [x0, x ; F ])(x0 − x )k, k(I − [x0, x ; F ])(x0 − x )k) (21) ∗ ∗ ≤ ω0(αkx0 − x k, βkx0 − x k)

≤ ω0(αr0, βr0) < 1.

By (21) and the Banach perturbation lemma [2, 3], we deduce that A0 is invertible and −1 0 ∗ 1 kA0 F (x )k ≤ ∗ ∗ . (22) 1 − ω0(αkx0 − x k, βkx0 − x k) We can write by method (1) that

∗ ∗ −1 y0 − x = x0 − x − A F (x0) 0 (23) −1 0 ∗ 0 ∗ −1 ∗ ∗ = A0 F (x )F (x ) ([u0, v0; F ] − [x0, x ; F ])kx0 − x k.

In view of (7), (8) (for i = 1), (13), (15), (22) and (23), we get in turn that

∗ −1 0 ∗ 0 ∗ −1 ∗ ∗ ky0 − x k ≤ kA0 F (x )kkF (x ) ([u0, v0; F ] − [x0, x ; F ])kkx0 − x k ∗ ∗ ω1(ku0 − x0k, kv0 − x k)kx0 − x k ≤ ∗ ∗ 1 − ω0(αkx0 − x k, βkx0 − x k) ω1(kF (x0)k, kF (x0)k) ≤ ∗ ∗ 1 − ω0(αkx0 − x k, βkx0 − x k) ∗ ∗ ∗ ∗ ω1(k[x0, x ; F ](x0 − x )k, k[x0, x ; F ](x0 − x )k) (24) ≤ ∗ ∗ 1 − ω0(αkx0 − x k, βkx0 − x k) ∗ ∗ ∗ ∗ ω1(v0(kx0 − x k)kx0 − x k, v0(kx0 − x k)kx0 − x k) ≤ ∗ ∗ 1 − ω0(αkx0 − x k, βkx0 − x k) ∗ ∗ ∗ = g1(kx0 − x k)kx0 − x k ≤ kx0 − x k < r,

∗ which shows (18) for n = 0 and y0 ∈ B(x , r), where we also used

∗ ∗ ku0 − x k = kx0 − x + F (x0)k ∗ ∗ = k(I + [x0, x ; F ])(x0 − x )k ∗ ∗ ≤ kI + [x0, x ; F ]kkx0 − x k ≤ αr

∗ ∗ ∗ ∗ and kv0 − x k ≤ kI − [x0, x ; F ]kx0 − x k ≤ βr so u0, v0 ∈ B(x , r) (by (17)). Next, we must show B0 is invertible, which will make z0 well deﬁned by the second substep of method (1) for n = 0. Using (3), (7), (9), (12), (14) and (24), we get in turn that [46] Ioannis K. Argyros and Santhosh George

0 ∗ −1 0 ∗ kF (x ) (B0 − F (x ))k 0 ∗ −1 0 ∗ 0 ∗ −1 ≤ kF (x ) ([y0, x0; F ] − F (x ))k + kF (x ) ([y0, x0; F ] − [u0, v0; F ])k ∗ ∗ ≤ ω0(ky0 − x k, kx0 − x k) + ω(ky0 − u0k, kx0 − v0k) ∗ ∗ ∗ ≤ ω0(g1(kx0 − x k)kx0 − x k, kx0 − x k) ∗ ∗ + ω(ky0 − x k + ku0 − x k, kF (x0)k) ∗ ∗ ∗ ≤ ω0(g1(kx0 − x k)kx0 − x k, kx0 − x k) ∗ ∗ ∗ ∗ + ω(g1(kx0 − x k + α)kx0 − x k, ϕ0(kx0 − x k)kx0 − x k) ∗ = p(kx0 − x k) ≤ p(r) < 1, so B0 is invertible and

−1 0 ∗ 1 kB0 F (x )k ≤ ∗ . (25) 1 − p(kx0 − x k)

It follows that z0 and x1 are well deﬁned by method (2). Then, by the second substep of method (1) for n = 0, (1), (8) (for i = 2), (16), (24) and (25), we have in turn that

∗ ∗ −1 0 ∗ 0 ∗ −1 kz0 − x k ≤ ky0 − x k + kB0 F (x )kkF (x ) F (y0)k ∗ ϕ(ky0 − x k) ∗ ≤ 1 + ∗ ky0 − x k 1 − p(kx0 − x k) ∗ ∗ ϕ(g1(kx0 − x k)kx0 − x k) ∗ ∗ ≤ 1 + ∗ g1(kx0 − x k)kx0 − x k 1 − p(kx0 − x k) ∗ ∗ ∗ = g2(kx0 − x k)kx0 − x k ≤ kx0 − x k < r,

∗ so (19) holds for n = 0 and z0 ∈ B(x , r). Then, from the last substep of method (1) for n = 0, (8) (for i = 3) and (25), we get in turn that

∗ ∗ −1 0 ∗ 0 ∗ −1 kx1 − x k ≤ kz0 − x k + kB0 F (x )kkF (x ) F (z0)k ∗ ϕ(kz0 − x k) ∗ ≤ 1 + ∗ kz0 − x k 1 − p(kx0 − x k) ∗ ∗ ϕ(g2(kx0 − x k)kx0 − x k) ∗ ∗ ≤ 1 + ∗ g2(kx0 − x k)kx0 − x k 1 − p(kx0 − x k) ∗ ∗ ∗ = g3(kx0 − x k)kx0 − x k ≤ kx0 − x k < r,

∗ which shows (20) and x1 ∈ B(x , r). The induction for (18)–(20) is completed in an analogous way, if we replace x0, y0, u0, v0, z0, x1 by xk, yk, uk, vk, zk, xk+1, respectively in the previous estimates. Then, from the estimate

∗ ∗ kxk+1 − x k ≤ ckxk − x k < r,

∗ where c = g3(kx0 − x ∗ k) ∈ [0, 1), we deduce that limk→∞ xk = x and xk+1 ∈ ∗ ∗ ∗ B(x , r). The uniqueness part is shown by assuming y ∈ Ω1 with F (y ) = 0. Local convergence of a multi-step high order method [47]

Deﬁne linear operator T by T = [y∗, x∗; F ]. Using (12) and (21), we have in turn that 0 ∗ −1 0 ∗ ∗ ∗ kF (x ) (T − F (x ))k ≤ ω0(0, ky − x k) ≤ ω0(0,R1) < 1, so T is invertible. If then follows from the identity 0 = F (y∗)−F (x∗) = T (y∗ −x∗) that x∗ = y∗.

Remark 2.2 Method (1) is not changing if we use the new instead of the old conditions [10, 11]. Moreover, for the error bounds in practice we can use the computational order of convergence (COC) [22]

∗ kxn+2−x k ln ∗ kxn+1−x k ξ = ∗ for each n = 1, 2,... kxn+1−x k ln ∗ kxn−x k or the approximate computational order of convergence (ACOC)

ln kxn+2−xn+1k ξ∗ = kxn+1−xnk for each n = 0, 1, 2,... ln kxn+1−xnk kxn−xn−1k instead of the error bounds obtained in Theorem 2.1.

3. Numerical Examples

The numerical examples are presented in this section. We choose Z 1 [x, y; F ] = F 0(y + θ(x − y)) dθ. 0 In the ﬁrst example, we compute the convergence radius and (COC) not given in [12].

Example 3.1 3 ¯ ∗ T Let B1 = B2 = R , Ω = U(0, 1), x = (0, 0, 0) . Deﬁne function F on D for w = (x, y, z)T by e − 1 T F (w) = ex − 1, y2 + y, z . 2 Then, x∗ = (0, 0, 0)T and the Fréchet-derivative is given by

ex 0 0 0 F (v) =  0 (e − 1)y + 1 0 . 0 0 1

L0 Ls+L0t Notice that using the (10) conditions, we get ω0(s, t) = 2 (s+t), ω1(s, t) = 2 , 1 1 1 1 L0 1 L0 ω(s, t) = 2 L(s+t), ϕ0(t) = ϕ(t) = 2 (1+e ), α = β = 1+ 2 (1+e ), L0 = e−1, L = e. The parameters are

r1 = 0.1524, r2 = 0.7499, r3 = 0.0578 = r, ξ = 4.9984. [48] Ioannis K. Argyros and Santhosh George

The work in [10, 11, 12] cannot be used in the next example, since B1 = B2 6= Ri. This example is also used to show how to compute the convergence radii in abstract space setting.

Example 3.2 ∗ Let B1 = B2 = C[0, 1], Ω = U¯(x , 1) and consider the nonlinear integral equation of the mixed Hammerstein-type [4, 6, 20] deﬁned by

Z 1 x(t)2 x(s) = K(s, t) dt, 0 2 where the kernel K is the Green’s function deﬁned on the interval [0, 1] × [0, 1] by ( (1 − s)t, t ≤ s, K(s, t) = s(1 − t), s ≤ t.

The solution x∗(s) = 0 is the same as the solution of equation (2), where F : C[0, 1] → C[0, 1]) is deﬁned by

Z 1 x(t)2 F (x)(s) = x(s) − K(s, t) dt. 0 2 Notice that Z 1 1 K(s, t) dt ≤ . 0 8 Then, we have that

Z 1 F 0(x)y(s) = y(s) − K(s, t)x(t) dt, 0 and F 0(x∗(s)) = I, 1 kF 0(x∗)−1(F 0(x) − F 0(y))k ≤ kx − yk. 8

t+s 9 We can choose ω0(t, s) = ω1(t, s) = ω(s, t) = 16 , ϕ0(t) = ϕ(t) = 16 and α = β = 25 16 . The parameters are

r1 = 0.6124, r2 = 0.1898, r3 = 0.1214 = r.

References

[1] Amat, Sergio, and Miguel A. Hernández, and Natalia Romero. "Semilocal convergence of a sixth order iterative method for quadratic equations." Appl. Numer. Math. 62, no. 7 (2012): 833–841. Cited on 42. [2] Argyros, Ioannis K. Computational theory of iterative methods. Vol. 15 of Studies in Computational Mathematics. Amsterdam: Elsevier B. V., 2007. Cited on 42 and 45. Local convergence of a multi-step high order method [49]

[3] Argyros, Ioannis K. "A semilocal convergence analysis for directional Newton methods." Math. Comp. 80, no. 273 (2011): 327–343. Cited on 42 and 45. [4] Argyros, Ioannis K., and Saïd Hilout. "Weaker conditions for the convergence of Newton’s method." J. Complexity 28, no. 3 (2012): 364–387. Cited on 42 and 48. [5] Argyros, Ioannis K., and Saïd Hilout. Computational methods in nonlinear analysis. Efficient algorithms, fixed point theory and applications. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2013. Cited on 42. [6] Cordero, Alicia, and Eulalia Martínez, and Juan R. Torregrosa. "Iterative methods of order four and five for systems of nonlinear equations." J. Comput. Appl. Math. 231, no. 2 (2009): 541–551. Cited on 42 and 48. [7] Cordero, Alicia et al. "A modified Newton-Jarratt’s composition." Numer. Algo- rithms 55, no. 1 (2010): 87–99. Cited on 42. [8] Cordero, Alicia, and Juan R. Torregrosa, and María P. Vassileva. "Increasing the order of convergence of iterative schemes for solving nonlinear systems." J. Comput. Appl. Math. 252 (2013): 86–94. Cited on 42. [9] Ezquerro, J.A., and M.A. Hernández, and A.N. Romero. "Aproximación de solu- ciones de algunas ecuaciones integrales de Hammerstein mediante métodos itera- tivos tipo Newton." In XXI Congreso de Ecuaciones Diferenciales y Aplicaciones, XI Congreso de Matemática Aplicada, Ciudad Real, 21-25 septiembre 2009, 1-8. Universidad de Castilla-La Mancha, 2009. Cited on 42. [10] Grau, Miquel, and José Luis Díaz-Barrero. "An improvement of the Euler- Chebyshev iterative method." J. Math. Anal. Appl. 315, no. 1 (2006): 1–7. Cited on 42, 47 and 48. [11] Grau-Sánchez, Miquel, and Ángela Grau, and Miquel Noguera. "Ostrowski type methods for solving systems of nonlinear equations." Appl. Math. Comput. 218, no. 6 (2011): 2377–2385. Cited on 42, 47 and 48. [12] Grau-Sánchez, Miquel, and Miquel Noguera, and Sergio Amat. "On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods." J. Comput. Appl. Math. 237, no. 1 (2013): 363–372. Cited on 41, 42, 47 and 48. [13] Gutiérrez, José Manuel, and Ángel A. Magreñán, and Natalia Romero. "On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions." Appl. Math. Comput. 221 (2013): 79–88. Cited on 42. [14] Kantorovich, Leonid V., and Gleb P. Akilov. Functional analysis. Second edition. Oxford-Elmsford, N.Y.: Pergamon Press, 1982. Cited on 42. [15] Magreñán, Ángel A. "Different anomalies in a Jarratt family of iterative root- finding methods." Appl. Math. Comput. 233 (2014): 29–38. Cited on 42. [16] Magreñán, Ángel A. "A new tool to study real dynamics: the convergence plane." Appl. Math. Comput. 248 (2014): 215–224. Cited on 42. [17] Petković, Miodrag S.et al. Multipoint methods for solving nonlinear equations. Amsterdam: Elsevier/Academic Press, 2013. Cited on 42. [18] Ren, Hongmin, and Ioannis K. Argyros. "Improved local analysis for a certain class of iterative methods with cubic convergence." Numer. Algorithms 59, no. 4 (2012): 505–521. Cited on 42. [50] Ioannis K. Argyros and Santhosh George

[19] Rheinboldt, Werner C. "An adaptive continuation process for solving systems of nonlinear equations." In Mathematical models and numerical methods, 129–142. Warsaw: Banach Center Publ. 3, 1978. Cited on 42. [20] Sharma, Janak Raj, and Puneet Gupta. "An eﬃcient ﬁfth order method for solving systems of nonlinear equations." Comput. Math. Appl. 67, no. 3 (2014): 591–601. Cited on 42 and 48. [21] Traub, Joseph F. Iterative methods for the solution of equations. New-York: AMS Chelsea Publishing, 1982. Cited on 42. [22] Weerakoon, S., and T.G.I. Fernando. "A variant of Newton’s method with accel- erated third-order convergence." Appl. Math. Lett. 13, no. 8 (2000): 87–93. Cited on 42 and 47.

Ioannis K. Argyros Department of Mathematical Sciences Cameron University Lawton, OK 73505 USA E-mail: [email protected]

Santhosh George Department of Mathematical and Computational Sciences NIT Karnataka India-575 025 India E-mail: [email protected]

Received: February 2, 2017; ﬁnal version: June 8, 2017; available online: August 28, 2017. Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 51-59 DOI: 10.1515/aupcsm-2017-0004

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Joanna Jureczko Strong sequences and partition relations Communicated by Tomasz Szemberg

Abstract. The ﬁrst result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdös-Rado theorem (1956). On the other hand in 60’s of the last century Eﬁmov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdös-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.

1. Introduction

Let α, β and λ be cardinals and n < ω. The arrow notation

n (α) → (β)λ

n S denotes the following partition relation: if [α] = i<α Pi, then there are A ⊂ α n and i < λ such that |A| = β and [A] ⊂ Pi, (see e.g. [2]). Partition calculus has been considered by many researchers. Below there are presented some famous results in this topic of partition relations, (followed by [1]). The ﬁrst result in this topic is given by Ramsey (1930) who proved that

2 U → (U)2 for non-principal ultraﬁltres on ω. Such ultraﬁltres were called "Ramsey" by Galvin around 1968. Other results belong to Rowbottom (1971) and Ketonen (1973).

AMS (2010) Subject Classiﬁcation: 03E02, 03E05, 03E20, 54A25. Keywords and phrases: Strong sequences, Erdös-Rado theorem, partition relations, inaccessible cardinals, singular cardinals. [52] Joanna Jureczko

There are results for ultraﬁltres on uncountable measurable cardinals. Moreover, Sierpiński (1933) proved that ℵ0 2 2 6→ (ℵ1)2. Probably the most famous theorem in partition calculus belongs to Erdös-Rado (1956) who proved that for every n - natural number and β - cardinal there is

+ + n+1 expn(β) → (β )β . The existence of diﬀerent kinds of partition relations is still explored. On the other hand a combinatorial method, so called strong sequences method, was introduced by Eﬁmov in the 60s’ of the last century in the subbase of the Cantor cube, which was used for proving some well known theorems in dyadic spaces, i.e., continuous images of the Cantor cube, see [3]. The problem of the existence and consequences of the existence of strong sequences in other spaces was explored in 90’s of the last century, see [8] and [9] for some historical details. The main goal of this paper is to show that the generalized versions on strong sequences, see [10], and Erdös-Rado theorem, see [5], are equivalent for regular and singular cardinals. There will be also shown some exceptions for which this equivalence does not hold.

2. Notations and deﬁnitions

Let (X, r) be a set with relation r and let κ be a cardinal. Let a, b, c ∈ X. We say that a and b have a bound iﬀ there exists c such that

(a, c) ∈ r and (b, c) ∈ r.

We say that a set A ⊂ X is a κ-directed iﬀ every subset of A of cardinality less than κ has a bound.

Definition 2.1 ([10]) A sequence (Hφ)φ<α, Hφ ⊂ X is called a κ-strong sequence if

(1) Hφ is κ-directed for all φ < α,

(2) Hψ ∪ Hφ is not κ-directed for all φ < ψ < α. Notice that in Deﬁnition 2.1 we do not really think about a sequence but rather about the family of sets with properties (1) and (2), so we can index (Hφ)φ<α not by ordinal but by cardinal numbers.

Definition 2.2 ([13]) A pair (F,G) of functions, where Φ is an ordinal.

G:Φ → 2X and F : X → 2Φ with the property that for any α, β if α < β, then there exists a ∈ G(α) such that β ∈ F (a) is called twin functions. Strong sequences and partition relations [53]

Definition 2.3 ([13]) A map g : K → X, where K ⊂ Φ such that (1) g(α) ∈ G(α) for any α ∈ K,

(2) for any α, β ∈ K if α < β, then β ∈ F (g(β)) is called a selector of twin functions (F,G). All other notations are standard in the ﬁeld and can ﬁnd in [2, 4, 7].

3. Theorems

In this part we present generalizations of three theorems: on strong sequences (compare [10]), on twin functions (compare [13]) and Erdös-Rado theorem (compare [5, 2]). Theorem 3.1 is proved in [10], Theorem 3.2 is a generalized version of Theorem on twin functions in [13]. Theorem 3.3 has been proved in [2], pp.8-11. Let β and η be cardinals. By β η we denote: η is β-strong inaccessible, i.e., β < η and αλ < η, whenever α < β, λ < η.

Theorem 3.1 (on κ-strong sequences) Let β, µ, η be cardinals such that ω ≤ β η, µ < β and β, η be regular. Let X be a set of cardinality η. If there exists a κ-strong sequences {Hα ⊂ X : α < η} with µ |Hα| ≤ 2 for all α < η, then there exists a κ-strong sequence {Tα ⊂ X : α < β} with |Tα| < κ for all α < β and κ < η.

Theorem 3.2 (on twin functions) Let β, µ, η be cardinals such that ω ≤ β η, µ < β and β, η be regular. Let X be a set of cardinality η. If there exists a pair (F,G) of twin functions G: η → 2X , F : X → 2η such that |G(α)| ≤ 2µ for all α < η, then there exists a selector g : K → X of twin functions with |K| = β.

Theorem 3.3 (Erdös-Rado type) Let β, µ, η be cardinals such that ω ≤ β η, µ < β and β, η be regular. Let X be a set of cardinality η. Then

2 (η) → (β)µ.

4. Proofs

In this part all proofs needed for showing equivalence of theorems from previous part will be presented. We introduce the following notation SS means theorem on strong sequence, TF means theorem on twin functions, ER means Erdös-Rado type theorem. [54] Joanna Jureczko

The structure of this section is as follows

SS =⇒ TF ⇑ ⇓ TF ⇐= ER

Proof. SS =⇒ TF Take a pair (F,G) of twin functions such that |G(α)| ≤ 2µ. For any α < η consider

−1 Hα = G(α) \ F (α).

Notice that {Hα : α < η} is a κ-strong sequence for κ < η. If not, then by Defini- tion 2.1 the set Hα ∪ Hβ would be an κ-directed set for α < β. By Definition 2.2 −1 there exists a ∈ Hα such that β ∈ F (a). Hence a ∈ F (β), which contradicts to definition of Hβ. By Theorem 3.1 there exists a κ-strong sequence {Tα : α < β} such that |Tα| < κ for any α < β. Consider a function g : β → X such that g(α) ∈ Tα for α < β. Obviously g(α) ∈ Hα ⊂ G(α), hence condition (1) of Definition 2.3 is fulfilled. Let a ∈ g(α) ∈ G(α). By Definition 2.3 we have γ ∈ F (a) = F (g(α)) for α < γ. Hence condition (2) of Definition 2.3 of twin functions is fulfilled.

Proof. TF =⇒ ER Without the loss of generality we can assume that X = {xα : α < η}. Let

f :[X]2 → λ be a function which determines a partition

2 [X] = {Aγ : γ < λ}.

We will show that for some Aγ of cardinality β there is f|Aγ = const. Deﬁne functions

G: η → 2X such that

G(α) = {xα ∈ X : {xα, xξ} ∈ Aγ for some γ < λ and any α < ξ}, F : X → 2η such that

F (xα) = {ξ < κ : {xα, xξ} ∈ Aγ for some γ < λ and α < ξ}.

We will show that functions in a pair (F,G) are twin functions. Take α < ξ. Let xα ∈ G(α), then {xα, xξ} ∈ Aγ for some α < λ. Hence ξ ∈ F (xα). By Theorem 3.2 there exists a selector of twin functions g : K → X with K ⊂ κ such that |K| = β. This means that for each α, ξ ∈ K we have 2 {xα, xξ} ∈ Aγ for some α < ξ. Suppose that [K] 6⊂ Aγ for any γ < λ. Let

{xα, xξ} ∈ Aρ for α < ξ and {xγ , xδ} ∈ Aσ for γ < δ Strong sequences and partition relations [55] and ρ 6= σ. By Deﬁnition 2.3 we have respectively

g(α) = xα and hence ξ ∈ F (g(α)) and g(γ) = xγ and hence δ ∈ F (g(γ)).

If ξ < δ, then ξ ∈ F (g(α)) and then {xα, xξ}, {xγ , xδ} ∈ Aρ. If δ < ξ, then 2 ξ ∈ F (g(γ)) and then {xα, xξ}, {xγ , xδ} ∈ Aσ. A contradiction. Hence [K] ⊂ Aγ for some γ < λ. Proof. ER =⇒ TF Without the loss of generality we can assume that X = {xα : α < η}. Assume that (F,G) is a pair G: η → 2X , F : X → 2η of twin function. We can assume that for any β < η G(ξ) = {xγ (ξ): γ < λ}. Let P :[X]2 → λ be a function such that

if P ({xα, xξ}) = γ, then α ∈ F (xγ (ξ)).

By Theorem 3.3 there exists a set K ⊂ X of cardinality β such that [K]2 ⊂ P −1(γ) for some γ ≤ λ. By above information, α ∈ F (xγ (ξ)) for all α, ξ ∈ K with α < ξ. Then g(ξ) = xγ (ξ) for ξ ∈ K is a selector of twin functions. Proof. TF =⇒ SS Assume that there exists a κ-strong sequence {Hα ⊂ X : α < η} for κ < η such µ that |Hα| ≤ 2 . Consider the following set

Aα = {T ⊂ Hα : |Tα| < ω and T ∪ Hξ is not κ-directed for any ξ > α}.

By Deﬁnition 2.1 the set Aα is non-empty for any α < η. Let

X = {T : T ∈ Aα for some α < η}.

Deﬁne a pair of functions (F,G):

X G: η → 2 such that G(α) = Aα, η F : X → 2 such that F (T ) = {ξ < η : T ∪ Hξ is not κ-directed}.

We will show that (F,G) are twin functions. Let α < ξ. Then for some T ∈ G(α) the set T ∪ Hξ is not κ-directed. Hence ξ ∈ F (T ). By Theorem 3.2 there exists a selector of twin functions g : K → X , where |K| = β. By (1) in Deﬁnition 2.3 the sets g(α) are κ-directed for any α ∈ K, by (2) in Deﬁnition 2.3 the sets g(α) ∪ Hξ are not κ-directed whenever α < ξ. Hence {g(α): α ∈ K} is a required κ-strong sequence. [56] Joanna Jureczko

5. Some counterexamples

We show that such strong assumptions of cardinals in Theorem 3.1 are necessary because if we omit them we can construct the following counterexamples.

Example 5.1 S Let µ = ℵ1, η = ℵ2 and X = ω2. Let X = {Hα : α < ω2} with |Hα| = ℵ1 be a partition of X. Let G be a graph on X such that for each β < α < ω2 there is eβ,α between Hβ and Hα such that the endvertices of {eβ,α : β < α} in Hα are distinct. Define the relation r: for finite S ⊂ X there is b with (a, b) ∈ r, a ∈ S iff S is bounded in Z. This gives a system {Hα : α < ω2} which is an ω-strong sequence assumed in the theorem. By Theorem 3.1 there exists an ω-strong sequence {Tα : α < ω1} consisting of finite sets. Define the set with using the following function f : ω1 → P (ω1) given by the formula η ∈ f(ξ) iff there are x ∈ Tξ, y ∈ Tη, y < x with {y, x} ∈ Z. Then each f(ξ) is finite and for ξ < η < ω1 either ξ ∈ f(η) or η ∈ f(ξ) holds, but this contradicts Hajnal Set Mapping Theorem, (see [6]).

Example 5.2 ℵ0 ℵ2 Let µ = ℵ1 and η = 2 . Let D , where D = {0, 1} denote the generalized Cantor ℵ2 discontinuum. For each α < ℵ2 consider the projection πα : D → {0, 1} and let −1 ℵ2 B = {πα : α < ℵ2, i ∈ {0, 1}}. The family B forms a subbase in D . Take the base generated by B as a set X and consider inclusion as a relation on X. The ℵ2 ℵ2 ℵ2 weight and the character of D are equal ℵ2. Let λ: 2 → D be a bijection. Consider a family Hα = {x ∈ X : λ(α) ∈ x}. Hence the family is centered because T ℵ2 Hα = λ(α) and of cardinality ℵ2 and obviously {Hα : α < 2 } forms an ω- strong sequence. By Theorem 3.1 there exists an ω-strong sequence {Tα : α < ℵ1} T consisting of ﬁnite sets. Then the family { Tα : α < ℵ1} is consisting of open ℵ2 pairwise disjoint sets. It means that cellularity of D is not less than ℵ1. But from Marczewski Theorem on cellularity of generalized Cantor discontinua, (see [11]), we have that the cellularity is ℵ0. A contradiction.

6. Applications and conclusions for regulars

In this section some applications and conclusions of theorems from Section 3 will be presented. The next theorem is the well known theorem proved by Erdös- Rado in [5], (see also [2]).

Corollary 6.1 (Erdös-Rado theorem) For any inﬁnite cardinal λ the following statement is true

λ + + 2 (2 ) → (λ )λ.

Proof. We have (2λ)λ = 2λ, thus λ+ (2λ)+ for all λ ≥ ω.

The next two corollaries were proved with using Theorem 3.1 on strong sequences in [12]. Strong sequences and partition relations [57]

Corollary 6.2 If X is a regular topological space, then d(X) ≤ χ(X)c(X).

Corollary 6.3 If X is a regular topological space, then w(X) ≤ χ(X)c(X). For completeness this paper we prove two corollaries using Theorem 3.1 on strong sequences. Corollary 6.4 If X, Y are topological spaces. Then c(X × Y ) ≤ 2c(X)+c(Y ). Proof. Let λ + X = {Uα × Vα : α(2 ) } be a family of open non-empty sets and let c(X × Y ) ≤ 2λ. We will show that c(X) + c(Y ) = λ. Let M0 ⊂ X be a maximal family of pairwise disjoint sets. Let M0 be the ﬁrst element of an ω-strong sequence. Suppose that for β < α < (2λ)+ the ω-strong S sequence {Mβ : β < α} such that Mβ ⊂ X \ γ<β Mγ is a maximal family λ of pairwise disjoint sets has been deﬁned. Obviously |Mβ| ≤ 2 for all β < α. S S Hence X\ β<α Mβ is non-empty. Let Mα ⊂ X \ β<α Mβ be a maximal family of pairwise disjoint sets. Thus we have obtained the ω-strong sequence λ + λ + + {Mα : α < (2 ) }. By Theorem 3.1 applying for η = (2 ) , β = λ , µ = λ and + κ = ω there exists an ω-strong sequence {Tα : α < λ } such that |Tα| < ω for all α < λ+. Hence c(X) = λ+ or c(Y ) = λ+. A contradiction.

Corollary 6.5 If X is a Hausdorff space then |X| ≤ 2χ(X)+c(X). Proof. Let λ = χ(X)+c(X). Assume that |X| > 2λ. Without the loss of generality λ + we can assume that X = {xα : α < (2 ) }. Let x0 ∈ X be an arbitrary element, B0 ⊂ X be a maximal local base in point x0 and let B0 be the first element in a strong sequence. Suppose that for β < α < (2λ)+ the ω-strong sequence S {Bβ : β < α} such that Bβ ⊂ X \ γ<β Bγ is a maximal local base in point S λ xβ ∈ X \ Bγ has been defined. Obviously |Bβ| ≤ 2 for all β < α. Hence S γ<β S X\ Bβ is non-empty. Let xα ∈ X \ Bβ be an arbitrary element. Let β<α S β<α Bα ⊂ X \ β<α Bβ be a maximal local base in point xα. Thus we have obtained λ + the ω-strong sequence {Bα : α < (2 ) }. By Theorem 3.1 applying for η = (2λ)+, β = λ+, µ = λ and κ = ω there + + exists an ω-strong sequence {Tα : α < λ } such that |Tα| < ω for all α < λ . By Definition 2.1 there exists a family of open non-empty pairwise disjoint sets + + {Uα : α < λ } such that Uα ∈ Tα. Hence c(X) = λ . A contradiction. [58] Joanna Jureczko

7. Results for singulars

In [2] p.8, the following result for singulars was proved. Since the proof of Theorem 7.1 is short we cite it here, (see also [2]). Theorem 7.1 Let β and η be cardinals such that ω ≤ β η with η-regular and β-singular. Then + 2 (κ) → (β )β. Proof. For β-singular we have β+ η. By Theorem 3.3 (with β and µ replaced by β+ and β respectively) we obtain our claim. Using the similar arguments as in Section 4 we easy obtain that Theorem 7.1 is also equivalent to both following results. Theorem 7.2 Let β and η be cardinals such that ω ≤ β η with η-regular and β-singular. Let X be a set of cardinality η. If there exists a κ-strong sequences {Hα ⊂ X : α < η} β with |Hα| ≤ 2 for all α < η, then there exists a κ-strong sequence {Tα ⊂ X : + + α < β } with |Tα| < κ for all α < β and κ < η. Theorem 7.3 Let β and η be cardinals such that ω ≤ β η with η-regular and β-singular. Let X be a set of cardinality η. If there exists a pair (F,G) of twin functions G: η → 2X , F : X → 2η such that |G(α)| ≤ 2β for all α < η, then there exists a selector g : K → X of twin functions with |K| = β+. Notice that the assumption of regularity of η cannot be omitted. The next theorem says about it. Theorem 7.4 Let β and η be cardinals such that ω ≤ β η and µ < β be cardinals with η-singular and β-regular. Then 2 η 6→ (β)µ. Proof. Suppose that the theorem is not true. Let f :[η]2 → µ be a function which determines a partition 2 [η] = {Aξ : ξ < µ}.

Notice that at least one Aξ has cardinality greater than β. If not, then

[ X Aξ = Aξ < µ · β < η. ξ<µ ξ<µ A contradiction. Consider a function g : µ → β such that g(α) = |f −1(α)| for any α < µ. Since at least one Aξ has cardinality greater than β, then sup{g(α): α < µ} > β, but β is regular. A contradiction.

Corollary 7.5 If λ is an inﬁnite cardinal, then λ + 2 2 6→ (λ )2. Strong sequences and partition relations [59]

References

[1] Baumgartner, James E., and Alan D. Taylor, and Stanley Wagon. Structural properties of ideals. Vol 197 of Dissertationes Math. (Rozprawy Mat.). Warszawa: PWN–Polish Scientific Publishers, 1982. Cited on 51. [2] Comfort, W. Wistar, and Stylianos A. Negrepontis. Chain conditions in topology. Vol 79 of Cambridge Tracts in Mathematics. Cambridge-New York: Cambridge University Press, 1982. Cited on 51, 53, 56 and 58. [3] Efimov, B. A. "Dyadic bicompacta." (Russian) Trudy Moskov. Mat. Obšč. 14 (1965): 211–247. Cited on 52. [4] Engelking, Ryszard. General topology. Vol. 60 of Monografie Matematyczne. Warszawa: PWN–Polish Scientific Publishers, 1977. Cited on 53. [5] Erdös, Paul, and Richard Rado. "A partition calculus in set theory." Bull. Amer. Math. Soc. 62 (1956): 427–489. Cited on 52, 53 and 56. [6] Hajnal, András. "Proof of a conjecture of S. Ruziewicz." Fund. Math. 50 (1961/1962): 123–128. Cited on 56. [7] Jech, Thomas. Set theory. New York-London: Academic Press, 1978. Cited on 53. [8] Jureczko, Joanna. "On inequalities among some cardinal invariants." Math. Aeterna 6, no. 1 (2016): 87–98 Cited on 52. [9] Jureczko, Joanna. "Strong sequences and independent sets." Math. Aeterna 6, no. 2 (2016): 141–152. Cited on 52. [10] Jureczko, Joanna. "κ-strong sequences and the existence of generalized independent families." Open Mathematics (2017) (to appear). Cited on 52 and 53. [11] Szpilrajn, Edward. "Remarque sur les produits cartésiens d’espaces topologiques." C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 (1941): 525–527. Cited on 56. [12] Turzański, Marian. "Strong sequences and the weight of regular spaces." Comment. Math. Univ. Carolin. 33, no. 3 (1992): 557–561. Cited on 56. [13] Turzański, Marian. "On the selector of twin functions." Comment. Math. Univ. Carolin. 39, no. 2 (1998): 303–307. Cited on 52 and 53.

Wrocław University of Science and Technology Wrocław Poland E-mail: [email protected]

Received: February 28, 2016; ﬁnal version: August 26, 2017; available online: October 2, 2017.

Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 61-72 DOI: 10.1515/aupcsm-2017-0005

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Akbar Rezaei, Arsham Borumand Saeid and Andrzej Walendziak Some results on pseudo-Q algebras Communicated by Justyna Szpond

Abstract. The notions of a dual pseudo-Q algebra and a dual pseudo-QC algebra are introduced. The properties and characterizations of them are investigated. Conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra are given. Commutative dual pseudo-QC algebras are considered. The interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.

1. Introduction

G. Georgescu and A. Iorgulescu [6] and independently J. Rachůnek [15], introduced pseudo-MV algebras which are a non-commutative generalization of MV- algebras. After pseudo-MV algebras, pseudo-BL algebras [7] and pseudo-BCK algebras [8] were introduced and studied by G. Georgescu and A. Iorgulescu. A. Wal- endziak [18] gave a system of axioms defining pseudo-BCK algebras. W.A. Dudek and Y.B. Jun defined pseudo-BCI algebras as an extension of BCI-algebras [5]. Y.H. Kim and K.S. So [11] discussed on minimal elements in pseudo-BCI algebras. G. Dymek studied p-semisimple pseudo-BCI algebras and then defined and investigated periodic pseudo-BCI algebras [3]. A. Walendziak [19] introduced pseudo-BCH algebras as an extension of BCH- algebras and studied ideals in such algebras. The notion of BE-algebras was introduced by H.S. Kim and Y.H. Kim [10]. B.L. Meng [13] introduced the notion of CI-algebras as a generalization of BE-algebras and dual BCK/BCI/BCH-algebras. R.A. Borzooei et al. defined and studied pseudo-BE algebras which are a generalization of BE-algebras [1]. A. Rezaei et al. introduced the notion of pseudo-CI algebras as a generalization

AMS (2010) Subject Classification: 06F35, 03G25, 03B52. Keywords and phrases: Pseudo-BCI/BCK algebra, (dual) pseudo-BCH algebra, dual pseudo- Q/QC algebra, (commutative) pseudo-CI/BE algebra. [62] A. Rezaei, A. Borumand Saeid and A. Walendziak of pseudo-BE algebras and proved that the class of commutative pseudo-CI algebras coincides with the class of commutative pseudo-BCK algebras [16]. Recently, Y.B. Jun et al. defined and investigated pseudo-Q algebras [9] as a generalization of Q-algebras [14]. In this paper, we define dual pseudo-Q and dual pseudo-QC algebras. We investigate the properties and characterizations of them. Moreover, we provide some conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra. We also consider commutative dual pseudo-QC algebras and prove that the class of such algebras coincides with the class of commutative pseudo-BCI algebras. Finally, the interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.

2. Preliminaries

In this section, we review the basic deﬁnitions and some elementary aspects that are necessary for this paper.

Definition 2.1 ([5]) An algebra X = (X; →, , 1) of type (2, 2, 0) is called a pseudo-BCI algebra if it satisﬁes the following axioms: for all x, y, z ∈ X,

(psBCI1)( x → y) ((y → z) (x → z)) = 1,

(psBCI2)( x y) → ((y z) → (x z)) = 1,

(psBCI3) x → ((x → y) y) = 1 and x ((x y) → y) = 1,

(psBCI4) x → x = x x = 1,

(psBCI5) x → y = y x = 1 =⇒ x = y,

(psBCI6) x → y = 1 ⇐⇒ x y = 1. Every pseudo-BCI algebra X satisfying, for every x ∈ X, condition

(psBCK) x → 1 = 1 is said to be a pseudo-BCK algebra ([12]). From [4] it follows that a pseudo-BCI-algebra X = (X; →, , 1) has the following property (for all x, y ∈ X)

(psEx) x → (y z) = y (x → z). Definition 2.2 ([17]) A (dual) pseudo-BCH algebra is an algebra (X; →, , 1) of type (2, 2, 0) verifying the axioms (psBCI4)–(psBCI6) and (psEx).

Remark 2.3 Obviously, every pseudo-BCI algebra is a pseudo-BCH algebra. Some results on pseudo-Q algebras [63]

Definition 2.4 ([16]) An algebra X = (X; →, , 1) of type (2, 2, 0) is called a pseudo-CI algebra if, for all x, y, z ∈ X, it satisﬁes the following axioms:

(psCI1) x → x = x x = 1,

(psCI2)1 → x = 1 x = x,

(psCI3) x → (y z) = y (x → z),

(psCI4) x → y = 1 ⇐⇒ x y = 1. Remark 2.5 Since every pseudo-BCH algebra satisﬁes (psCI1)–(psCI4), pseudo-BCH algebras are contained in the class of pseudo-CI algebras.

A pseudo-CI algebra X = (X; →, , 1) verifying condition

(psBE) x → 1 = x 1 = 1, for all x ∈ X, is said to be a pseudo-BE algebra (see [1]).

Proposition 2.6 ([2]) Any pseudo-BCK algebra is a pseudo-BE algebra. In a pseudo-CI algebra X we can introduce a binary relation “ ≤ ” by

x ≤ y ⇐⇒ x → y = 1 ⇐⇒ x y = 1 for all x, y ∈ X.

An algebra X = (X; →, , 1) of type (2, 2, 0) is called commutative if for all x, y ∈ X, it satisﬁes the following identities:

(i)( x → y) y = (y → x) x, (ii)( x y) → y = (y x) → x. From [2] (see Theorem 3.4) it follows that any commutative pseudo-BE algebra is a pseudo-BCK algebra. By Theorem 3.9 of [16], any commutative pseudo-CI algebra is a pseudo-BE algebra. Therefore we obtain

Proposition 2.7 Commutative pseudo-CI algebras coincide with commutative pseudo-BE algebras and with commutative pseudo-BCK algebras (hence also coincide with commutative pseudo-BCI algebras and with commutative pseudo-BCH algebras).

Definition 2.8 ([9]) An algebra X = (X; ∗, , 0) of type (2, 2, 0) is called a pseudo-Q algebra if, for all x, y, z ∈ X, it satisﬁes the following axioms:

(psQ1) x ∗ x = x x = 0,

(psQ2) x ∗ 0 = x 0 = x,

(psQ3)( x ∗ y) z = (x z) ∗ y. [64] A. Rezaei, A. Borumand Saeid and A. Walendziak

3. Dual pseudo-Q algebras

Definition 3.1 An algebra X = (X; →, , 1) of type (2, 2, 0) is called a dual pseudo-Q algebra if, for all x, y, z ∈ X, it veriﬁes the following axioms:

(dpsQ1) x → x = x x = 1,

(dpsQ2)1 → x = 1 x = x,

(dpsQ3) x → (y z) = y (x → z).

In a dual pseudo-Q algebra, we can introduce two binary relations ≤→ and

≤ by x ≤→ y ⇐⇒ x → y = 1 and x ≤ y ⇐⇒ x y = 1. Proposition 3.2 Let X = (X; →, , 1) be a dual pseudo-Q algebra. Then X is a pseudo-CI algebra if and only if ≤→ = ≤ . Example 3.3 (i) Let X = {1, a, b, c, d}. Deﬁne binary operations → and on X by the following tables ([16]):

→ 1 a b c d 1 a b c d 1 1 a b c d 1 1 a b c d a 1 1 c c 1 a 1 1 b c 1 and . b 1 d 1 1 d b 1 d 1 1 d c 1 d 1 1 d c 1 d 1 1 d d 1 1 c c 1 d 1 1 b c 1

Then X = (X; →, , 1) is a dual pseudo-Q algebra which is not a pseudo- BCI algebra, since b 6= c and b → c = c b = 1 (that is, (psBCI5) does not hold in X).

(ii) Let X = {1, a, b, c}. Deﬁne binary operations → and on X by the following tables: → 1 a b c 1 a b c 1 1 a b c 1 1 a b c a 1 1 b c and a 1 1 c c . b 1 1 1 1 b 1 1 1 c c 1 1 a 1 c 1 1 c 1

Then X = (X; →, , 1) is a dual pseudo-Q algebra which is not a pseudo-CI algebra, because b → c = 1 but b c = c. By deﬁnition, we have

Proposition 3.4 Any pseudo-CI algebra is a dual pseudo-Q algebra. Some results on pseudo-Q algebras [65]

Remark 3.5 The converse of Proposition 3.4 does not hold. See Example 3.3 (ii). Proposition 3.6 Let X be a dual pseudo-Q algebra. If one of the following identities: (1)( y → x) → x = y → x,

(2)( y → x) x = y x, (3)( y x) → x = y → x, (4)( y x) x = y → x, (5)( y x) x = y x, (6)( y x) → x = y x, (7)( y → x) x = y → x, (8)( y → x) → x = y x holds in X, then X is a trivial algebra.

Proof. Suppose, for example, that (1) is satisﬁed. Let x ∈ X. Applying (dpsQ1), (1) and (dpsQ2) we have 1 = x → x = (x → x) → x = 1 → x = x. Thus X is a trivial algebra.

Proposition 3.7 Let X be a dual pseudo-Q algebra. If one of the following identities: (1)( y → x) → x = x → y,

(2)( y → x) x = x y, (3)( y x) → x = x → y, (4)( y x) x = x → y, (5)( y x) x = x y, (6)( y x) → x = x y, (7)( y → x) x = x → y, (8)( y → x) → x = x y holds in X, then X is a trivial algebra. Proof. The proof is similar to the proof of Proposition 3.6.

Proposition 3.8 In a dual pseudo-Q algebra X, for all x, y, z ∈ X, we have:

(1) if 1 ≤→ x or 1 ≤ x, then x = 1, [66] A. Rezaei, A. Borumand Saeid and A. Walendziak

(2) x ≤ y → z ⇐⇒ y ≤→ x z, (3) x → 1 = x 1,

(4)( x → y) → 1 = (x → 1) (y 1) and (x y) 1 = (x 1) → (y → 1),

(5) if x ≤→ y, then x → 1 = y → 1,

(6) if x ≤ y, then x 1 = y 1, (7) y → ((y → x) x) = 1 and y ((y x) → x) = 1,

Proof. (1) Let 1 ≤→ x. Then 1 → x = 1. Now, by (dpsQ2) we obtain x = 1.

Similarly, if 1 ≤ x, then x = 1. (2) Let x, y, z ∈ X. By (dpsQ3),

x (y → z) = 1 ⇐⇒ y → (x z) = 1. Consequently, (2) holds. (3) We have x → 1 = x → (x x) = x (x → x) = x 1. (4) Let x, y, z ∈ X. Then

(x → y) → 1 = (x → y) → [(x → 1) (x → 1)] = (x → 1) [(x → y) → (x → 1)] = (x → 1) [(x → y) → (x → (y y))] = (x → 1) [(x → y) → (y (x → y))] = (x → 1) [y ((x → y) → (x → y))] = (x → 1) (y 1). The proof of the second part is similar.

(5) Let x ≤→ y. Then x → y = 1 and so y → 1 = y 1 = y (x → y) = x → (y y) = x → 1. Thus y → 1 = x → 1. (6) The proof is similar to the proof of (5).

(7) By (dpsQ3) and (dpsQ1) we get

y → ((y → x) x) = (y → x) (y → x) = 1 and y ((y x) → x) = (y x) → (y x) = 1.

A dual pseudo-Q algebra X = (X; →, , 1) satisfying the conditions (psBCI1) and (psBCI2) is said to be a dual pseudo-QC algebra. The following example shows that there exist pseudo-Q algebras which do not satisfy (psBCI1) or (psBCI2). Some results on pseudo-Q algebras [67]

Example 3.9

(i) Dual pseudo-Q algebra from Example 3.3 (ii) satisﬁes (psBCI2) but it does not satisfy (psBCI1), since

(a → b) ((b → c) (a → c)) = b (1 c) = c 6= 1.

(ii) Let X = {1, a, b, c, d, e, f, g, h}. We deﬁne the binary operations → and on X as follows ([17]):

→ 1 a b c d e f g h 1 a b c d e f g h 1 1 a b c d e f g h 1 1 a b c d e f g h a 1 1 1 1 d e f g h a 1 1 1 1 d e f g h b 1 c 1 1 d e f g h b 1 c 1 1 d e f g h c 1 c b 1 d e f g h c 1 c b 1 d e f g h and . d d d d d 1 g h e f d d d d d 1 h g f e e e e e e h 1 g f d e e e e e g 1 h d f f f f f f g h 1 d e f f f f f h g 1 e d g h h h h e f d 1 g g h h h h f d e 1 g h g g g g f d e h 1 h g g g g e f d h 1

Then X = (X; →, , 1) is a dual pseudo-Q algebra which does not satisfy (psBCI1) and (psBCI2). Indeed,

(c → a) ((a → b) (c → b)) = c (1 b) = c b = b 6= 1

and

(c a) → ((a b) → (c b)) = c → (1 → b) = c → b = b 6= 1.

(iii) Let X = {1, a, b, c}. Deﬁne binary operations → and on Xby the following tables: → 1 a b c 1 a b c 1 1 a b c 1 1 a b c a 1 1 b b and a 1 1 b c . b 1 a 1 c b 1 a 1 a c 1 1 1 1 c 1 1 1 1

Then X = (X; →, , 1) is a dual pseudo-QC algebra. Lemma 3.10 Let X = (X; →, , 1) be a dual pseudo-QC algebra and x, y ∈ X. Then x → y = 1 if and only if x y = 1.

Proof. Let x → y = 1. Using (dpsQ2) and (psBCI1) we obtain

x y = x (1 y) = (1 → x) ((x → y) (1 → y)) = 1.

Similarly, if x y = 1, then x → y = 1. From Lemma 3.10 we have [68] A. Rezaei, A. Borumand Saeid and A. Walendziak

Proposition 3.11 Any dual pseudo-QC algebra is a pseudo-CI algebra. Remark 3.12 The converse of Proposition 3.11 does not hold. See Example 3.9 (ii). Proposition 3.13 Every pseudo-BCI algebra is a dual pseudo-QC algebra.

Proof. Let X be a pseudo-BCI algebra. It is easy to see that X satisﬁes (dpsQ1)– (dpsQ3), that is, it is a dual pseudo-Q algebra. Moreover, X obviously satisﬁes (psBCI1) and (psBCI2). Consequently, X is a dual pseudo-QC algebra. Remark 3.14

In a dual pseudo-QC algebra, ≤→=≤ . Set ≤=≤→ (=≤ ). Proposition 3.15 Let X be a dual pseudo-QC algebra and x, y, z ∈ X. Then: (1) if x ≤ y, then y → z ≤ x → z and y z ≤ x z, (2) if x ≤ y, then z → x ≤ z → y and z x ≤ z y.

Proof. (1) Let x ≤ y. Then x → y = 1. By (dpsQ2) and (psBCI1) we have

(y → z) (x → z) = 1 ((y → z) (x → z)) = (x → y) ((y → z) (x → z)) = 1. Hence y → z ≤ x → z. The proof of the second part is similar.

(2) Let x ≤ y. Hence x → y = 1. Applying (dpsQ2) and (psBCI1) we obtain

(z → x) → (z → y) = 1 ((z → x) → (z → y)) = (x → y) ((z → x) → (z → y)) = (z → x) → ((x → y) (z → y)) = 1.

Hence z → x ≤ z → y. Similarly, z x ≤ z y. Theorem 3.16 Let X be a dual pseudo-Q algebra. Then X is a pseudo-QC algebra if and only if it satisﬁes the following implications:

(∗) y ≤→ z =⇒ x → y ≤ x → z,

(∗∗) y ≤ z =⇒ x y ≤→ x z. Proof. If X is a pseudo-QC algebra, then it satisﬁes (∗) and (∗∗) by Proposi- tion 3.15. Conversely, suppose that implications (∗) and (∗∗) hold for all x, y, z ∈

X. By Proposition 3.8 (7), y ≤→ (y → z) z. Using (∗) we get x → y ≤ x → ((y → z) z). Hence (x → y) (x → ((y → z) z)) = 1. Applying (psEx) we obtain (x → y) ((y → z) (x → z)) = 1, that is, (psBCI1) holds. Similarly, using (∗∗) we have (psBCI2). Some results on pseudo-Q algebras [69]

Proposition 3.17 Let X be a dual pseudo-QC algebra. Then X is a pseudo-BCI algebra if and only if it veriﬁes (psBCI5).

Proof. Let X be a dual pseudo-QC algebra satisfying (psBCI5). Clearly, X veriﬁes (psBCI1), (psBCI2) and (psBCI4). The axiom (psBCI3) follows from Proposi- tion 3.8 (7). By Lemma 3.10, (psBCI6) holds in X. Therefore, X is a pseudo-BCI algebra. The converse is obvious.

Proposition 3.18 Let X be a dual pseudo-QC algebra and x, y, z ∈ X such that x ≤ y and y ≤ z. Then x ≤ z.

Proof. Applying (dpsQ2) and (psBCI1) we get

x → z = 1 (x → z) = 1 (1 (x → z)) = (x → y) ((y → z) (x → z)) = 1, and therefore x ≤ z.

Corollary 3.19 If a dual pseudo-QC algebra X satisﬁes the condition (psBCI5), then (X; ≤) is a poset.

Theorem 3.20 If X is a commutative dual pseudo-QC algebra, then it is a pseudo-BCI algebra.

Proof. It is suﬃcient to prove that (psBCI5) holds in X. Let x, y ∈ X and x → y = y x = 1. Then

x = 1 → x = (y x) → x = (x y) → y = 1 → y = y.

Therefore, X satisﬁes (psBCI5). Thus X is a pseudo-BCI algebra. From Theorem 3.20 it follows

Corollary 3.21 Commutative dual pseudo-QC algebras coincide with commutative pseudo-BCI algebras.

4. Conclusion

Denote by psBCK, psBCI, psBCH, psCI, psBE, dpsQ, and dpsQC the classes of pseudo-BCK, pseudo-BCI, pseudo-BCH, pseudo-CI, pseudo-BE, dual pseudo-Q, and dual pseudo-QC algebras respectively. By deﬁnition, psBCK ⊂ psBCI and psBE ⊂ psCI ⊂ dpsQ. From Remarks 2.3 and 2.5 we obtain psBCI [70] A. Rezaei, A. Borumand Saeid and A. Walendziak

⊂ psBCH ⊂ psCI. Moreover, that psBCI ⊂ dpsQC ⊂ psCI follows from Propositions 3.13 and 3.11. By Proposition 2.7 and Corollary 3.21, commutative pseudo-QC algebras coincide with commutative algebras pseudo-BCK, -BCI, -BCH, -CI, -BE. Now, in the following diagram we summarize the results of this paper and the previous results in this ﬁled. An arrow indicates proper inclusion, that is, if X and Y are classes of algebras, then X → Y denotes X ⊂ Y. The mark X →C Y means that every commutative algebra of X belongs to Y.

dpsQ 6

C - psBEpsCI 6 ¡¡ ¡ C ¡¡ ¡ ¡ ? ¡¡ psBCH ¡ C ¡ ¡¡ 6 ¡ C C ¡ ¡¡ ¡ ? ¡ C - ¡ psBCIdpsQC ¡ ¡¡ 6 ¡¡ C ¡¡ ¡¡ ?¡ ¡¡ psBCK

Problem 4.1 Is it true that every commutative dual pseudo-Q algebra is a pseudo-BCK algebra?

Acknowledgment

The authors wish to express their sincere thanks to the referees for the valuable suggestions which lead to an improvement of this paper.

References

[1] Borzooei, Rajab Ali et al. "On pseudo BE-algebras." Discuss. Math. Gen. Algebra Appl. 33, no. 1 (2013): 95–108. Cited on 61 and 63. [2] Ciungu, Lavinia Corina. "Commutative pseudo BE-algebras." Iran. J. Fuzzy Syst. 13, no. 1 (2016): 131–144. Cited on 63. Some results on pseudo-Q algebras [71]

[3] Dymek, Grzegorz. "p-semisimple pseudo-BCI-algebras." J. Mult.-Valued Logic Soft Comput. 19, no. 5-6 (2012): 461–474. Cited on 61. [4] Dymek, Grzegorz. "On a period of elements of pseudo-BCI-algebras." Discuss. Math. Gen. Algebra Appl. 35, no. 1 (2015): 21–31. Cited on 62. [5] Dudek, Wieslaw A., and Young-Bae Jun. "Pseudo-BCI algebras." East Asian Math. J. 24, no. 2 (2008): 187–190. Cited on 61 and 62. [6] Georgescu, George, and Afrodita Iorgulescu. "Pseudo-MV algebras." Mult.-Valued Log. 6, no. 1-2 (2001): 95–135. Cited on 61. [7] Georgescu, George, and Afrodita Iorgulescu. "Pseudo-BL algebras: anoncommu- tative extension of BL algebras." In Abstracts of the Fifth International Conference FSTA 2000, Liptovský Ján, The Slovak Republic. January 31 - February 4, 2000, 90–92. Cited on 61. [8] Georgescu, George, and Afrodita Iorgulescu. "Pseudo-BCK algebras: an extension of BCK algebras." In Combinatorics, computability and logic: Proceedings of the Third International Conference on Combinatorics, Computability and Logic, edited by C.S Calude et al. Discrete Math. Theor. Comput. Sci., 97–114. London: Springer, 2001. Cited on 61. [9] Jun, Young Bae, and Hee Sik Kim, and Sun Shin Ahn. "Structures of pseudo ideal and pseudo atom in a pseudo Q-algebra." Kyungpook Math. J. 56, no. 1 (2016): 95–106. Cited on 62 and 63. [10] Kim, Hee Sik, and Young Hee Kim. "On BE-algebras." Sci. Math. Jpn. 66, no. 1 (2007): 113–116. Cited on 61. [11] Kim, Young Hee, and Keum Sook So. "On minimality in pseudo-BCI-algebras." Commun. Korean Math. Soc. 27, no. 1 (2012): 7–13. Cited on 61. [12] Kühr, Jan. "Pseudo BCK-semilattices." Demonstratio Math. 40, no. 3 (2007): 495–516. Cited on 62. [13] Meng, Biao Long. "CI-algebras." Sci. Math. Jpn. 71, no. 1 (2010) 11–17. Cited on 61. [14] Neggers, Joseph, and Sun Shin Ahn, and Hee Sik Kim. "On Q-algebras." Int. J. Math. Math. Sci. 27, no. 12 (2001): 749–757. Cited on 62. [15] Rachůnek, Jiří. "A non-commutative generalization of MV-algebras." Czechoslovak Math. J. 52(127), no. 2 (2002): 255–273. Cited on 61. [16] Rezaei, Akbar, and Arsham Borumand Saeid, and K. Youseﬁ Sikari Saber. "On pseudo-CI algebras." (submitted). Cited on 62, 63 and 64. [17] Rezaei, Akbar, and Arsham Borumand Saeid, and Andrzej Walendziak. "On pointed pseudo-CI algebras."(submitted). Cited on 62 and 67. [18] Walendziak, Andrzej. "On axiom systems of pseudo-BCK algebras." Bull. Malays. Math. Sci. Soc. (2) 34, no. 2 (2011): 287–293. Cited on 61. [19] Walendziak, Andrzej. "Pseudo-BCH-algebras." Discuss. Math. Gen. Algebra Appl. 35, no. 1 (2015): 5–19. Cited on 61. [72] A. Rezaei, A. Borumand Saeid and A. Walendziak

Akbar Rezaei Department of Mathematics Payame Noor University p.o.box. 19395-3697 Tehran Iran E-mail: [email protected]

Arsham Borumand Saeid Department of Pure Mathematics Faculty of Mathematics and Computer Shahid Bahonar University of Kerman Kerman Iran E-mail: [email protected]

Andrzej Walendziak Siedlce University, Faculty of Science Institute of Mathematics and Physic 3 Maja 54 Siedlce Poland E-mail: [email protected]

Received: March 10, 2017; ﬁnal version: August 30, 2017; available online: November 2, 2017. Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 73-76 DOI: 10.1515/aupcsm-2017-0006

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Raghavendra G. Kulkarni Intersect a quartic to extract its roots Communicated by Justyna Szpond

Abstract. In this note we present a new method for determining the roots of a quartic polynomial, wherein the curve of the given quartic polynomial is intersected by the curve of a quadratic polynomial (which has two unknown coeﬃcients) at its root point; so the root satisﬁes both the quartic and the quadratic equations. Elimination of the root term from the two equations leads to an expression in the two unknowns of quadratic polynomial. In addition, we introduce another expression in one unknown, which leads to determination of the two unknowns and subsequently the roots of quartic polynomial.

1. Introduction

This note presents a new method to determine the roots of a quartic polynomial. In this method, the curve of the given quartic polynomial is intersected by the curve of a quadratic polynomial (which has two unknown coeﬃcients) at its root point, say (r, 0). Hence the root, r, satisﬁes both the quartic and the quadratic equations. Elimination of r from the two equations leads to an expression in the two unknowns of quadratic polynomial. In addition, we introduce another expression of our choice in one unknown, to enable us to determine the two unknowns from the two expressions; and subsequently the roots of quartic polynomial are determined. The method is explained in detail as below.

AMS (2010) Subject Classiﬁcation: 12E05, 12E12. Keywords and phrases: Intersection of curves, quartic polynomial, quadratic polynomial, roots, resolvent cubic equation. [74] Raghavendra G. Kulkarni

2. Intersection of polynomials

Without loss of any generality, we consider here a depressed quartic polynomial (which has no x3 term),

p(x) = x4 + ax2 + bx + c, (1) where the coeﬃcients, a, b and c, are real and the roots of p(x) are in the complex ﬁeld. Our aim is to determine these roots using the method of intersection proposed here. The other methods of obtaining roots of quartic polynomials are mentioned in [1, 2]. Let us consider a quadratic polynomial,

q(x) = x2 − dx − f, (2) where the coeﬃcients, d and f, are unknowns (d, f ∈ C). We stipulate that the curve of the quadratic polynomial (2) intersects the curve of the given quartic polynomial (1) at its root, r. Therefore the root must satisfy the two equations, p(r) = 0 and q(r) = 0, which means:

r4 + ar2 + br + c = 0, (3)

r2 − dr − f = 0. (4) Eliminating r4, and then subsequently eliminating r2 from (3) using (4), and after some algebraic manipulations we get an expression for the root, r, in terms of d and f, r = −(d2f + f 2 + af + c)/(d3 + 2df + ad + b). (5) Eliminating r from (4) using (5) yields an expression in the two unknowns, d and f,

f 4 + 2af 3 + (ad2 − 3bd + a2 + 2c)f 2 − (bd3 − 4cd2 + abd + b2 − 2ac)f (6) + cd4 + acd2 + bcd + c2 = 0.

We need one more equation, which can facilitate the determination of two unknowns. To obtain such an equation, ﬁrst we rearrange (6) as,

[f + (a/2)]4 + [ad2 − 3bd + 2c − (a2/2)]f 2 − [bd3 − 4cd2 + abd + b2 + (a3/2) − 2ac]f + cd4 + acd2 + bcd + c2 − (a4/16) = 0.

Making a substitution, g = f + (a/2), in the above expression yields,

g4 + hg2 + jg + k = 0, (7) where h, j and k are given by:

h = ad2 − 3bd + 2c − (a2/2), j = −[bd3 + (a2 − 4c)d2 − 2abd + b2], Intersect a quartic to extract its roots [75]

k = cd4 + (ab/2)d3 + [(a3/4) − ac]d2 + [bc − (a2b/4)]d + c2 + (ab2/2) − (a2c/2) + (a4/16). Notice that setting j = 0 gives us the desired equation, which is a cubic equation in d, d3 + [(a2 − 4c)/b]d2 − 2ad + b = 0, (8) known as resolvent cubic equation. Solving (8) we determine three values of d. Also note that j = 0 makes (7) a quadratic equation in g2, g4 + hg2 + k = 0. (9) Solving (9), we determine g2 as, p g2 = (−h ± h2 − 4k)/2, leading to determination of four values of g, q p g = ± (−h ± h2 − 4k)/2. (10) Further using the relation, f = g − (a/2), four values of f are obtained from (10). Notice that for each value of d, we get four values of f. One can use any one of three values of d and corresponding four values of f to determine the four roots of given quartic polynomial (1) from the expression (5). However, if for a certain set of {d, f}, the denominator of (5) goes to zero, then such set has to be avoided and the other sets can be used to determine the roots of the quartic polynomial.

3. Numerical example

We solve one numerical example with the proposed method. Let the curve of the quartic polynomial, p(x) = x4 + 2x2 + 4x + 2, be intersected by the curve of a quadratic polynomial, q(x) = x2 − dx − f. The resolvent cubic equation (8) is obtained as, d3 − d2 − 4d + 4 = 0, and its three roots are obtained as, d = 1, 2, and −2. Choosing d = 1, the quartic equation (9) is obtained as: g4 − 8g2 + 25 = 0. Now, from (10) we determine four values of g as: 2.121320343559 + 0.7071067811865i, − 2.121320343559 − 0.7071067811865i, 2.121320343559 − 0.7071067811865i, − 2.121320343559 + 0.7071067811865i. Using the relation, f = g − (a/2), the corresponding four values of f are obtained as: 1.121320343559 + 0.7071067811865i, − 3.121320343559 − 0.7071067811865i, 1.121320343559 − 0.7071067811865i, − 3.121320343559 + 0.7071067811865i. We use (5) to determine the four roots of given quartic polynomial as: −0.7071067811865 − 0.2928932188134i, 0.7071067811865 − 1.707106781186i, −0.7071067811865 + 0.2928932188134i, 0.7071067811865 + 1.707106781186i. Use of other values of d to ﬁnd the roots of quartic polynomial is left as an exercise to the readers. [76] Raghavendra G. Kulkarni

4. Conclusions

We have presented a new method to obtain the roots of a quartic polynomial, wherein the curve of a quadratic polynomial with two unknown coeﬃcients intersects the curve of the quartic at a common root point (r, 0). So the root r satisﬁes both the quadratic and the quartic equations. Elimination of r from these equations leads to an expression in the two unknowns. Introduction of a cubic equation (known as resolvent cubic equation) in one of the unknowns leads to the determination of both unknowns, and further it results in the extraction of the roots of quartic polynomial.

Acknowledgment

The author is grateful to the anonymous referee, whose valuable comments resulted in improved manuscript. The author thanks the management of PES University, Bengaluru, for supporting this work.

References

[1] Dickson, Leonard E. First course in the theory of equations. New York: J. Wiley & sons, inc., 1922. Cited on 74. [2] Kulkarni, Raghavendra G. "Shifting the origin to solve quartic equations." The Mathematical Gazette 97, no. 539 (2013): 268–270. Cited on 74.

Department of Electronics & Communication Engineering PES University 100 Feet Ring Road, BSK III Stage Bengaluru - 560085 India E-mail: [email protected]; [email protected]

Received: September 25, 2017; ﬁnal version: November 4, 2017; available online: November 28, 2017. Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 77-93 DOI: 10.1515/aupcsm-2017-0007

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Sk. Nazmul On type-2 m-topological spaces Communicated by Tomasz Szemberg

Abstract. In the present paper, we deﬁne a notion of an m2-topological space by introducing a count of openness of a multiset (mset in short) and study the properties of m2-subspaces, mgp-maps etc. Decomposition theorems involving m-topologies and m2-topologies are established. The behaviour of the functional image and functional preimage of an m2-topologies, the continuity of the identity mapping and a constant mapping in m2-topologies are also examined.

1. Introduction

A classical set is a collection of objects where an object can occur only once. But there are a number of situations in science and real life where the repetition of an object is significant. Allowing repetition of elements, N. G. de Bruijn [4] first suggested to generalize classical sets to multisets (msets in short) in a private communication to D. E. Knuth. These sets are very useful structures arising in many areas of mathematics and computer science such as in prime factorization of integers, invariants of matrices in canonical form, zeros and poles of meromorphic functions, multicriteria decision making, knowledge representation in data based systems, biological systems membrane computing etc. Several researchers have worked in variety of terms viz. list, heap, bunch, bag, sample, weighted set, occurrence set and fireset used in different contexts but conveying synonymity with mset. Many authors like Yager [23], Miyamoto [17], Hickman [13], Blizard [3], Girish and John [6, 7], Hallez et al. [10] etc., have studied the set theoretic properties of msets. Some hybridizations of msets may be found in [1, 2, 12, 16]. Structural study, such as topological, are found in [8, 9, 20, 22], algebraic in

AMS (2010) Subject Classiﬁcation: 00A05, 03D70, 03E99, 06A99. Keywords and phrases: Multi sets; multi topologies; multi subspace; m2-topologies; mgp- maps. [78] Sk. Nazmul

[18, 19]. Note that the m-topology defined on msets by Girish and John [8, 9] as actually an ordinary set τ of some msets. In this paper, an attempt has been made in allowing the repetition of members of m-topology τ. A definition of type-2 m-topology is introduced which will be called m2-topology. The relevance of this approach in fuzzy setting have been done by A. Sostak˘ [21], M. S. Ying [24], U. Höhle and A. Sostak˘ [14], T. Kubiak [15], and Hazra, Chattopadhyay and Samanta [5, 11]. In brief, in this paper, we have defined a notion of an m2- topological space by introducing a count of openness of an mset, m2-cotopological space by introducing a count of closedness of an mset. Moreover, m2-subspaces, mgp-mappings and some of their important properties are studied. Decomposition theorems involving m-topologies and m2-topologies are established. The behaviour of functional image and functional preimage of an m2-topology, the continuity of the identity mapping and a constant mapping in m2-topologies are also examined.

2. Preliminaries

This section consists of some deﬁnitions and results of msets and m-topologies which will be used in the main works of the paper. Unless otherwise stated, X will be assumed to be an initial universal set and N represents the set of all non negative integers.

2.1. Multi sets (or msets)

Definition 2.1 ([7]) An mset M drawn from the universal set X is represented by a count function CM deﬁned as CM : X → N, where N represents the set of non negative integers. Here CM (x) is the number of occurrences of the element x in the mset M. The presentation of the mset M drawn from X = {x1, x2, . . . , xn} will be as M = {x1/m1, x2/m2, . . . , xn/mn}, where mi is the number of occurrences of the element w xi,i = 1, 2, . . . , n in the mset M. Also here for any positive integer w, [X] is the set of all msets whose elements are in X such that no element in the mset occurs more than w times and [X]∞ is the set of all msets whose elements are in X such that there is no limit on the number of occurrences of an element in an mset. As in [7], [X]w and [X]∞ will be referred to as mset spaces. MS(X) denotes the set of all msets drawn from X.

Definition 2.2 ([7]) Let M1 and M2 be two msets drawn from a set X. Then M1 is said to be submset of M2 if CM1 (x) ≤ CM2 (x) for all x ∈ X. This relation is denoted by M1 ⊆ M2.

Set M1 is said to be equal to M2 if CM1 (x) = CM2 (x) for all x ∈ X, which will be denoted by M1 = M2. On type-2 m-topological spaces [79]

Definition 2.3 ([7]) Let w be a positive integer and {Mi; i ∈ I} be a non-empty family of msets in [X]w. Then T (a) the intersection of the sets Mi, is a set denoted by i∈I Mi, such that ^ CT (x) = C (x) for all x ∈ X; Mi Mi i∈I i∈I

S (b) the union of the sets Mi, is a set denoted by i∈I Mi, such that _ CS (x) = C (x) for all x ∈ X; Mi Mi i∈I i∈I

w c (c) the complement of any mset Mi in [X] is a set denoted by Mi , such that C c (x) = w − C (x) for all x ∈ X. Mi Mi Definition 2.4 ([18]) Let X and Y be two non-empty sets and f : X → Y be a mapping. Then

(i) the image of an mset M ∈ [X]w under the mapping f is a set denoted by f(M), such that

( W −1 CM (x), if f (y) 6= φ, C (y) = f(x)=y f(M) 0, otherwise;

(ii) the inverse image of an mset N ∈ [Y ]w under the mapping f is a set denoted −1 by f (N), such that Cf −1(N)(x) = CN [f(x)].

Proposition 2.5 ([18]) Let X, Y and Z be three non-empty sets and f : X → Y , g : Y → Z be two w w mappings. If Mi ∈ [X] , Ni ∈ [Y ] , i ∈ I then

(i) M1 ⊆ M2 ⇒ f(M1) ⊆ f(M2); S S (ii) f i∈I Mi = i∈I f[Mi];

−1 −1 (iii) N1 ⊆ N2 ⇒ f (N1) ⊆ f (N2);

−1 S S −1 (iv) f i∈I Mi = i∈I f [Mi];

−1 T T −1 (v) f i∈I Mi = i∈I f [Mi];

−1 (vi) f(Mi) ⊆ Nj ⇒ Mi ⊆ f [Nj];

−1 −1 −1 (vii) g[f(Mi)] = [gf](Mi) and f [g (Nj)] = [gf] (Nj). [80] Sk. Nazmul

Proposition 2.6 ([18]) Let X and Y be two non-empty sets and f : X → Y be a mapping. If M ∈ [X]w and N ∈ [Y ]w, then (i) M ⊆ f −1[f(M)]; (ii) f −1[f(M)] = M, if f is injective; (iii) f[f −1(N)] ⊆ N; (iv) f[f −1(N)] = N, if f is surjective.

Definition 2.7 ([18])

Let P ⊆ X. Then for each n ∈ N, we deﬁne an mset nP over X, where CnP (x) = n for all x ∈ P . This msets are called level msets.

2.2. Msets topology Definition 2.8 ([8]) Let M ∈ [X]w be a multiset and P ∗(M) be the collection of all submsets of M. A subcollection τ of P ∗(M) is said to be a multiset topology (m-topology in short) on M if (i) M, ∅ ∈ τ; (ii) the intersection of any two msets in τ belongs to τ; (iii) the union of any number of msets in τ belongs to τ. The pair (M, τ) is called an m-topological space on M.

Definition 2.9 ([9]) Let (M, τ) be an m-topological space and N be a submset of M. The collection τN = {N ∩ U : U ∈ τ} is an m-topology on N, called a subspace m-topology. Definition 2.10 ([9]) Let M and N be two m-topological spaces. The mset function f : M → N is said to be continuous if for each open submset V of N, the mset f −1(V ) is an open submset of M, where f −1(V ) is the mset of all points x/m in M for which f(x/m) ∈ nV for some n.

3. m2-topological spaces

In this section, we introduce a count of openness, a count of closedness, m2- topological spaces, m2-cotopological spaces, m2-subspaces, mgp-maps and some of their important properties are studied. Decomposition theorems involving m- topologies and m2-topologies are established. The behaviour of the functional image and the functional preimage of an m2-topology, the continuity of the identity mapping and a constant mapping in m2-topologies are also examined. Unless otherwise stated, X denotes a non-empty set, w is a positive integer, N denotes the set of all non negative integers, Nw is the set of all non negative On type-2 m-topological spaces [81]

integers not greater than w and [X]w is the collection of all those msets whose elements are in X such that no element in the mset occurs more than w times. Definition 3.1 w A mapping τ :[X] → Nw is called a count of openness (CO) or an m2-topology on [X]w if it satisﬁes the following conditions:

(O1) τ(0X) = τ(wX) = w;

w (O2) τ(M1 ∩ M2) ≥ τ(M1) ∧ τ(M2) for M1,M2 ∈ [X] , S w (O3) τ i∈∆ Mi ≥ ∧i∈∆τ(Mi) for any Mi ∈ [X] , i ∈ ∆. The pair ([X]w, τ) is called an m2-topological space (m2ts). Example 3.2 w w Let τ0 :[X] → Nw, τw :[X] → Nw be two mappings deﬁned by τ0(0X) = w τ0(wX) = w, τ0(M) = 0 for all M ∈ [[X] − {0X, wX}] and τw(M) = w for all w w M ∈ [X] . Then τ0 and τw are two m2-topologies on [X] . Definition 3.3 w w A mapping F :[X] → Nw is called a count of closedness (CC) on [X] if it satisﬁes the following conditions:

(C1) F(0X) = F(wX) = w;

w (C2) F(M1 ∪ M2) ≥ F(M1) ∧ F(M2) for M1,M2 ∈ [X] ; T w (C3) F i∈∆ Mi ≥ ∧i∈∆F(Mi) for any Mi ∈ [X] , i ∈ ∆. The pair ([X]w, F) is called an m2-cotopological space. Proposition 3.4 Let τ and F be a count of openness and a count of closedness of [X]w, respectively. w c Then the mapping Fτ :[X] → Nw, deﬁned by Fτ (M) = τ(M ) is a count of closedness on [X]w. Proof. Let τ, F be a count of openness and a count of closedness of [X]w respec- w c tively and Fτ :[X] → Nw be a mapping deﬁned by Fτ (M) = τ(M ). Since c c (0X) = wX and (wX) = 0X, it follows that Fτ (0X) = τ(wX) = w and Fτ (wX) = τ(0X) = w. w Next let M1, M2 be any two members of [X] . Then

c c c Fτ (M1 ∪ M2) = τ([M1 ∪ M2] ) = τ(M1 ∩ M2 ) c c ≥ τ(M1 ) ∧ τ(M2 ) = Fτ (M1) ∧ Fτ (M2).

w Again let Mi, i ∈ ∆ be any collection of members of [X] . Then

T T c S c Fτ ( i∈∆ Mi) = τ [ i∈∆ Mi] = τ i∈∆ Mi c ≥ ∧i∈∆τ(Mi ) = ∧i∈∆Fτ (Mi).

w c Therefore, the mapping Fτ :[X] → Nw, deﬁned by Fτ (M) = τ(M ), is a count of closedness on [X]w. [82] Sk. Nazmul

Proposition 3.5 Let τ and F be a count of openness and a count of closedness of [X]w, respectively. w c Then the mapping τF :[X] → Nw deﬁned by τF (M) = F(M ) is a count of openness on [X]w.

Proof. Proof is similar to that of Proposition 3.4.

Proposition 3.6 Let τ and F be a count of openness and a count of closedness on [X]w, respectively.

Then τFτ = τ and FτF = F. Proof. Proof is straightforward.

Proposition 3.7 w Let τ1 and τ2 be two counts of openness on [X] . Then τ = τ1 ∩ τ2 deﬁned by w τ(M) = τ1(M) ∧ τ2(M) is a count of openness on [X] .

Proof. Clearly τ(0X) = τ(wX) = w. Next, let M1, M2 be any two members of [X]w. Then

τ(M1 ∩ M2) = τ1(M1 ∩ M2) ∧ τ2(M1 ∩ M2)

≥ [τ1(M1) ∧ τ1(M2)] ∧ [τ2(M1) ∧ τ2(M2)]

= [τ1 ∩ τ2](M1) ∧ [τ1 ∩ τ2](M2)

= τ(M1) ∧ τ(M2).

w Again let Mi, i ∈ ∆ be any collection of members of [X] . Then S T T τ i∈∆ Mi = τ1 i∈∆ Mi ∧ τ2 i∈∆ Mi ≥ [∧i∈∆τ1(Mi)] ∧ [∧i∈∆τ2(Mi)]

= ∧i∈∆[τ1 ∩ τ2](Mi)

= ∧i∈∆τ(Mi).

Therefore, τ = τ1 ∩ τ2, deﬁned by τ(M) = τ1(M) ∧ τ2(M), is a count of openness on [X]w.

Remark 3.8 w If {τi, i ∈ ∆} is any arbitrary family of counts of openness on [X] , then their T w intersection τ = i∈∆ τi, deﬁned by τ(M) = ∧i∈∆τi(M) for all M ∈ [X] , is a count of openness on [X]w.

Definition 3.9 w Let τ1 and τ2 be two counts of openness on [X] . Deﬁne

w τ1 ≤ τ2 iﬀ τ1(M) ≤ τ2(M) for all M ∈ [X] .

If τ1 ≤ τ2 then we say that τ1 is coarser or weaker or smaller than τ2 and τ2 is ﬁner or stronger or larger than τ1. On type-2 m-topological spaces [83]

Proposition 3.10 Let T be the collection of all counts of openness on [X]w. Then (T , ≤) is a complete lattice.

w Proof. Let τ0, τw be two counts of openness on [X] deﬁned in Example 3.2. Then τ0 ≤ τ ≤ τw for all τ ∈ T . Note that τ1 ∩ τ2 is the greatest lower bound (glb) of τ1 and τ2 for all τ1, τ2 ∈ T . T Moreover, {τ ∈ T : τ1 ≤ τ and τ2 ≤ τ} is the least upper bound (lub) of τ1 and τ2 for all τ1, τ2 ∈ T (we note that there exists at least one count of openness viz. τw, which is ﬁner than both τ1 and τ2). Therefore, (T , ≤) is a complete lattice.

Proposition 3.11 (First Decomposition Theorem) Let ([X]w, τ) be an m-topological space, where τ is a count of openness on [X]w. r w Then for each r(∈ Nw) ≤ w, τ = {M ∈ [X] : τ(M) ≥ r} is a multiset topology on wX.

Proof. Since τ(0X) = τ(wX) = w ≥ r, it follows that 0X, wX ∈ τr. Next let w r r r M1,M2 ∈ [X] be any two members of τ . Then τ (M1) ≥ r and τ (M2) ≥ r. w Since τ is a count of openness on [X] , it follows that τ(M1 ∩ M2) ≥ [τ(M1) ∧ r τ(M2)] ≥ r. Hence M1 ∩ M2 ∈ τ . w r Furthermore let, {Mi ∈ [X] , i ∈ ∆} be any collection of members of τ . Then S S r τ(Mi) ≥ r for all i ∈ ∆. So, τ i∈∆ Mi ≥ ∧i∈∆τ(Mi) ≥ r. Thus, i∈∆ Mi ∈ τ . Therefore, τr is an m-topology on wX.

Definition 3.12 r For each r(∈ Nw) ≤ w, the family τ , deﬁned in Proposition 3.11, is called the r-level m-topology on wX with respect to the count of openness τ.

Proposition 3.13 Let ([X]w, τ) be an m-topological space and {τ r : r ≤ w} be the family of all r-level m-topologies with respect to τ. Then this family is a descending family of m-topologies.

Proof. Let r ≥ s and M ∈ τ r. Then τ(M) ≥ r ≥ s, hence M ∈ τ s. Thus τ r ⊆ τ s and hence the family {τ r : r ≤ w} is descending family of m-topologies.

Definition 3.14 Let τ be a count of openness on [X]w. Then supp(τ) = {M ∈ [X]w : τ(M) > 0} is called the support set of τ.

It is clear that supp(τ) is an m-topology on wX.

Definition 3.15 w Let T be an m-topology on wX. Then a count of openness τ on [X] is said to be compatible with T if supp(τ) = T .

Proposition 3.16 Let T be an m-topology on wX. Then for each r ≤ w there exists a count of openness T r on [X]w compatible with T . [84] Sk. Nazmul

r w Proof. For each r ≤ w we deﬁne a mapping T :[X] → Nw by  w, if M ∈ {0X, wX}, r  T (M) = r, if M ∈ T − {0X, wX},  0, otherwise.

Then, clearly T r is a count of openness on [X]w compatible with T .

Proposition 3.17 (Second Decomposition Theorem) Let {Tr : r ≤ w} be a non-empty descending family of m-topologies on wX. Then w the mapping τ :[X] → Nw deﬁned by τ(M) = ∨{r ≤ w : M ∈ Tr} is a count of w r openness on [X] and Tr = τ holds for all r ≤ w.

Proof. Since 0X, wX ∈ Tr for all r ≤ w, it follows that τ(0X) = τ(wX) = w. Let w M1, M2 be any two members of [X] and let τ(Mi) =: ki for i = 1, 2. If ki = 0 for some i, then obviously τ(M1 ∩ M2) ≥ τ(M1) ∧ τ(M2). Assume now that k1 6= 0, k1 6= 0 and k = k1 ∧ k2. Since {Tr : r ≤ W } is a descending family of m-topologies, it follows that M1,M2 ∈ Tk and hence M1 ∩ M2 ∈ Tk. Thus

τ(M1 ∩ M2) = ∨{r ≤ w :(M1 ∩ M2) ∈ Tr} ≥ k

= k1 ∧ k2 = τ(M1) ∧ τ(M2).

w Moreover, let Mi for i ∈ ∆ be any collection of members of [X] and let τ(Mi) =: li for i ∈ ∆. If li = 0 for some i ∈ ∆, then obviously S τ i∈∆ Mi ≥ ∧i∈∆τ(Mi).

Now let li 6= 0 for all i ∈ ∆ and l = ∧i∈∆li. Since {Tr : r ≤ w} is a descending S family of m-topologies, it follows that Mi ∈ Tl, i ∈ ∆ and hence i∈∆ Mi ∈ Tl. Thus S S τ i∈∆ Mi = ∨ r ≤ w : i∈∆ Mi ∈ Tr ≥ l = ∧i∈∆li = ∧i∈∆τ(Mi).

w Therefore, the mapping τ :[X] → Nw deﬁned by τ(M) = ∨{r ≤ w : M ∈ Tr} is a count of openness on [X]w. For second part, let us assume ﬁrst that M ∈ Tr. Then τ(M) ≥ r and hence M ∈ τ r. Thus r Tr ⊆ τ . (1) Next let M ∈ τ r. Then τ(M) ≥ r this implies that there exists s ≥ r such that M ∈ Ts. Since {Tr : r ≤ w} is a descending family of m-topologies and s ≥ r, it follows that M ∈ Tr. Thus r τ ⊆ Tr. (2) r From (1) and (2), we have Tr = τ .

Proposition 3.18 w Let τ1 and τ2 be two counts of openness on [X] . Then τ1 = τ2 if and only if, r r τ1 = τ2 for all r ≤ w. On type-2 m-topological spaces [85]

w Proof. First let τ1 = τ2. Then τ1(M) = τ2(M) for all M ∈ [X] , so for each r w w r r r ≤ w, τ1 = {M ∈ [X] : τ1(M) ≥ r} = {M ∈ [X] : τ2(M) ≥ r}. Thus τ1 = τ2 for all r ≤ w. r r w Next let τ1 = τ2 for all r ≤ w. If τ1 6= τ2, then there exists an M ∈ [X] such s1+1 that τ1(M) 6= τ2(M). Let τ1(M) = s1, τ2(M) = s2 and s1 < s2. Then M ∈ τ2 s1+1 r r but M 6∈ τ1 , which contradicts our assumption that τ1 = τ2 for all r ≤ w. Therefore τ1 = τ2.

Proposition 3.19 r w Let T be an m-topology on wX. For each r ≤ w deﬁne a mapping T :[X] → Nw by  w, if M ∈ {0X, wX}, r  T (M) = r, if M ∈ T − {0X, wX},  0, otherwise.

r w r Then T a count of openness on [X] such that (T )r = T .

Proof. Proof follows from Proposition 3.16.

Definition 3.20 r Let T be an m-topology on wX. Then T , deﬁned in Proposition 3.19, is called w r an r-th count on wX and ([X] ,T ) is called an r-th count m2-topological spaces.

Definition 3.21 w w r r Let M ∈ [X] and let τ be a CO on [X] . Let τM := {M ∩ P : P ∈ τ } for r r ≤ w. Then {τM : r ≤ w} is a descending family of subspace m-topology on M.

Definition 3.22 w M w Let X be a non-empty set and M(6= 0X) ∈ [X] . A mapping τ :[X] → Nw is called a subspace m2-topology or a subspace count of openness (brieﬂy SCO) on M if it satisﬁes the following conditions:

M M (i) τ (M ∩ 0X) = τ (M ∩ wX) = w;

M Tn n M (ii) if M1,M2,...,Mn ⊆ M, then τ ( i=1 Mi) ≥ ∧i=1τ (Mi);

M Sn M (iii) if Mi ⊆ M, i ∈ ∆, then τ ( i∈∆ Mi) ≥ ∧i∈∆τ (Mi). The pair (M, τ M ) is called an m2-subspace of ([X]w, τ).

Proposition 3.23 w Let X be a non-empty set, τ be a CO on X and M(6= 0X) ∈ [X] . A mapping M w τ :[X] → Nw deﬁned by

∨{τ(Q): Q ∩ M = P,Q ∈ [X]w}, if P ⊆ M, τ M (P ) = 0, if P 6⊆ M is an SCO on M. [86] Sk. Nazmul

w Proof. From the fact that 0X, wX ∈ [X] and τ(0X) = τ(wX) = w, it follows that M M M τ (0X ∩ M) = τ(0X) = w and τ (wX ∩ M) ≥ τ(wX) = w. Also τ (wX) ≤ w. M Hence τ (wX ∩ M) = w. Tn Next let M1,M2,...,Mn ⊆ M and A = i=1 Mi. Let Ni be an arbitrary w member of [X] such that Ni ∩ M = Mi. Then Tn Tn Tn ( i=1 Ni) ∩ M = i=1(Ni ∩ M) = i=1 Mi = A.

Thus, M Tn n τ (A) ≥ τ( i=1 Ni) ≥ ∧i=1τ(Ni) and hence

M n w τ (A) ≥ ∨{Ni∈[X] : Ni∩M=Mi} ∧i=1 τ(Ni) n w = ∧i=1 ∨{Ni∈[X] : Ni∩M=Mi} τ(Ni) n M = ∧i=1τ (Mi).

S w Assume now that Mi ⊆ M for i ∈ ∆ and A = i∈∆ Mi. Let βi = {N ∈ [X] : N ∩ M = Mi}, i ∈ ∆. For any Ni ∈ βi, i ∈ ∆, we have S S S ( i∈∆ Ni) ∩ M = i∈∆(Ni ∩ M) = i∈∆ Mi = A, we also have M τ (Mi) = ∨{τ(N): N ∈ βi}, i ∈ ∆. M S S M S Thus τ ( i∈∆ Mi) ≥ τ( i∈∆ Ni) ≥ ∧i∈∆τ(Ni). Therefore τ ( i∈∆ Mi) ≥ M M ∧i∈∆τ (Mi) (similarly as above). Hence τ is an SCO on M.

Proposition 3.24 Let X, Y be two non-empty sets, f : X → Y be a mapping and τ be a CO on [X]w. w −1 w Then f(τ):[Y ] → Nw deﬁned by [f(τ)](N) = τ(f (N)), N ∈ [Y ] is a CO on [Y ]w.

−1 −1 Proof. Since f (0Y ) = 0X and f (wY ) = wX, it follows that

[f(τ)](0Y ) = [f(τ)](wY ) = w.

w Tn Now let N1,N2,...,Nn ∈ [Y ] and N = i=1 Ni. Then

−1 Tn −1 [f(τ)](N) = τ[f (N)] = τ[ i=1(f (Ni))] n −1 n ≥ ∧i=1τ[f (Ni)] = ∧i=1[f(τ)](Ni).

w S Finally, let Ni ∈ [Y ] , i ∈ ∆ and N = i∈∆ Ni. Then

−1 S −1 [f(τ)](N) = τ[f (N)] = τ[ i∈∆(f (Ni))] −1 ≥ ∧i∈∆τ[f (Ni)] = ∧i∈∆[f(τ)](Ni).

Therefore f(τ) is a CO on [Y ]w. On type-2 m-topological spaces [87]

Proposition 3.25 Let X, Y be two non-empty sets, f : X → Y be an onto mapping and ν be a CO on w −1 w −1 w [Y ] . Then f (ν):[X] → Nw deﬁned by [f (ν)](M) = ν[f(M)], M ∈ [X] is a CO on [X]w.

−1 Proof. From the fact f(0X) = 0Y , it follows that [f (ν)](0X) = w. AS f is −1 −1 onto, we have f (wY ) = wX and hence [f (ν)](wX) = w. Assume that w Tn M1,M2,...,Mn ∈ [X] and M = i=1 Mi. Then

−1 Tn [f (ν)](M) = ν[f(N)] = ν[ i=1(f(Mi))] n n −1 ≥ ∧i=1ν[f(Mi)] = ∧i=1[f (ν)](Mi).

w S Now let Mi ∈ [Y ] , i ∈ ∆ and M = i∈∆ Mi. Then

−1 S [f (ν)](M) = ν[f(M)] = ν[ i∈∆(f(Mi))] −1 ≥ ∧i∈∆ν[f(Mi)] = ∧i∈∆[f (ν)](Mi).

Therefore f −1(ν) is a CO on [X]w.

Proposition 3.26 Let X, Y be two non-empty sets, f : X → Y be an onto mapping and τ be a CO −1 on [X]w. Then [f(τ)]N (P ) = τ f (N)[f −1(P )] for all N ∈ [Y ]w and P ⊆ N.

Proof. Let N ∈ [Y ]w and P ⊆ N. Then

[f(τ)]N (P ) = ∨{[f(τ)](Q): Q ∩ N = P } = ∨{τ[f −1(Q)] : Q ∩ N = P } = ∨{τ[f −1(Q)] : f −1(Q ∩ N) = f −1(P )} (as f is onto) = ∨{τ[f −1(Q)] : f −1(Q) ∩ f −1(N) = f −1(P )} = ∨{τ[M]: M ∈ [X]w and M ∩ f −1(N) = f −1(P )} (as f is onto) −1 = τ f (N)[f −1(P )].

Proposition 3.27 Let X, Y be two non-empty sets, f : X → Y be a one-one mapping and ν be a CO on [Y ]w. Then [f −1(ν)]M (P ) ≤ νf(M)[f(P )] for all M ∈ [X]w and P ⊆ M.

Proof. Let M ∈ [X]w and P ⊆ M. Then

[f −1(ν)]M (P ) = ∨{[f −1(ν)](Q): Q ∩ M = P } = ∨{ν[f(Q)] : Q ∩ M = P } = ∨{ν[f(Q)] : f(Q ∩ M) = f(P )} (since f is one-one) = ∨{ν[f(Q)] : f(Q) ∩ f(M) = f(P )} ≤ ∨{ν[N]: N ∈ [Y ]w such that N ∩ f(M) = f(P )} = νf(M)[f(P )]. [88] Sk. Nazmul

Proposition 3.28 w w Let τ be a count of openness on [X] and A ⊆ X. If τA :[A] → Nw is a mapping deﬁned by w w τA(N) = ∨{τ(M): M ∈ [X] ,M ∩ wA = N},N ∈ [A] . w Then τA is a count of openness on [A] .

Proof. Since 0A = wA ∩ 0X, wA = wA ∩ wX and τ(0X) = τ(wX) = w, it follows that τA(0A) = τA(wA) = w. w w Let N1, N2 be any two members of [A] . Then we ﬁnd M1,M2 ∈ [X] such that N1 = M1 ∩w Y and N2 = M2 ∩ wA. Hence N1 ∩ N2 = (M1 ∩ M2) ∩ wA and τA(N1 ∩ N2) ≥ τ(M1 ∩ M2) ≥ τ(M1) ∧ τ(M2). Thus w τA(N1 ∩ N2) ≥ ∨{τ(M1) ∧ τ(M2): M1,M2 ∈ [X]

such that N1 = M1 ∩ wA, N2 = M2 ∩ wA}

≥ ∨{∨{τ(M1) ∧ τ(M2): M2 ∩ wA = N2} : M1 ∩ wA = N1}

= ∨{τ(M1) ∧ τA(N2): M1 ∩ wA = N1}

= τA(N1) ∧ τA(N2). w Now let Ni, i ∈ ∆ be any collection of members of [A] . Then we ﬁnd w S Mi ∈ [X] , i ∈ ∆ such that Ni = Mi ∩ wA, i ∈ ∆. It follows that Ni = S i∈∆ ( i∈∆ Mi) ∩ wA and similarly as above we have S S τA( i∈∆ Ni) ≥ τ( i∈∆ Mi) ≥ ∧i∈∆τ(Mi) ≥ ∧i∈∆τA(Ni). w Therefore, τA is a count of openness on [A] . Proposition 3.29 w w w Let ([A] , τA) be an m-subspace of the m-topological space ([X] , τ) and N ∈ [A] . Then w (i) FτA (N) = {Fτ (M): M ∈ [X] ,M ∩ wA = N};

(ii) if B ⊆ A ⊆ X, then τB = (τA)B. w w Proof. (i) Let ([A] , τA) be an m-subspace of the m-topological space ([X] , τ) and N ∈ [A]w. Then c w c FτA (N)τA(N ) = ∨{τ(M): M ∈ [X] ,M ∩ wA = N } c w c = ∨{τ(M): M ∈ [X] ,M ∩ wA = N} c c w c = ∨{Fτ (M ): M ∈ [X] ,M ∩ wA = N} w = ∨{Fτ (M): M ∈ [X] ,M ∩ wA = N}. (ii) Let P ∈ [B]w. Then w (τA)B(P ) = ∨{τA(N): N ∈ [A] ,N ∩ wB = P } w w = ∨{∨{τ(M): M ∈ [X] ,M ∩ wA = N} : N ∈ [A] ,N ∩ wB = P } w = ∨{τ(M): M ∈ [X] ,M ∩ wB = P } = τB(P ).

Therefore, if B ⊆ A ⊆ X, then τB = (τA)B. On type-2 m-topological spaces [89]

Definition 3.30 Let ([X]w, τ) and ([Y ]w, ν) be two m2-topological spaces and f : ([X]w, τ) → ([Y ]w, ν) be a mapping. Then f is called a count preserving map or an mgp-map if τ(f −1(N)) ≥ ν(N) for each N ∈ [Y ]w. Proposition 3.31 Let ([X]w, τ) be an m-topological space. Then the identity mapping f : ([X]w, τ) → ([X]w, τ) is an mgp-map. Proof. Since f is the identity mapping, it follows that f −1(N) = N for all N ∈ [X]w and hence τ(f −1(N)) = τ(N).

Remark 3.32 Let ([X]w, τ) and ([Y ]w, ν) be two m2-topological spaces and f : ([X]w, τ) → ([Y ]w, ν) be a constant mapping. Then f is not an mgp-map in general, which shows the following example. Example 3.33 w w Let X = {x, y, z}, Y = {a, b, c, d} and w = 3. Let τ :[X] → Nw, ν :[Y ] → Nw w be two mappings deﬁned by τ(0X) = τ(wX) = w, τ(M) = 0 for all M ∈ [[X] − w {0X, wX}] and ν(N) = w for all N ∈ [Y ] . Then τ and ν are two m2-topologies on [X]w, [Y ]w, respectively. Moreover, let f : ([X]w, τ) → ([Y ]w, ν) be a constant mapping, deﬁned by 3 −1 f(x) = a for all x ∈ X. If N = {a, a, b, c} ∈ [Y ] , then f (N) = 2X and −1 τ[f (N)] = τ(2X) = 0 6≥ 3 = w = ν(N). Therefore, the mapping f is not an mgp-map. Proposition 3.34 w w Let ([X] , τ) and ([Y ] , ν) be two m2-topological spaces. If τ(kX) = w for all k(≤ w) ∈ N, then the constant mapping f : ([X]w, τ) → ([Y ]w, ν) is an mgp-map.

Proof. Let f be a constant mapping and assume that there exists y0 ∈ Y such w −1 that f(x) = y0 for all x ∈ X. Then for any N ∈ [Y ] , f (N) = kX for some −1 k(≤ w) ∈ N. Hence τ(f (N)) = τ(kX) = w ≥ ν(N). Therefore, the constant mapping f : ([X]w, τ) → ([Y ]w, ν) is an mgp-map.

Proposition 3.35 Let ([X]w, τ) and ([Y ]w, ν) be two m2-topological spaces and let f : ([X]w, τ) → w ([Y ] , ν) be a mapping. Then f is a mgp-map iﬀ f :(wX, τr) → (wY, νr) is m- continuous for all r ≤ w. w w Proof. First let f : ([X] , τ) → ([Y ] , ν) be an mgp-map, r ≤ w and N ∈ νr. Then N ∈ [Y ]w and ν(N) ≥ r. Since f is an mgp-map, it follows that τ(f −1(N)) ≥ ν(N) ≥ r

−1 and hence f (N) ∈ τr. Therefore, f :(wX, τr) → (wY, νr) is m-continuous. Conversely, let f :(wX, τr) → (wY, νr) for all r ≤ w be m-continuous and N ∈ [Y ]w. Let moreover ν(N) = r. If r = 0, then obviously τ(f −1(N)) ≥ −1 −1 ν(N), otherwise N ∈ νr and hence f (N) ∈ τr. Thus, τ(f (N)) ≥ r = ν(N). Therefore, f : ([X]w, τ) → ([Y ]w, ν) is an mgp-map. [90] Sk. Nazmul

Proposition 3.36 0 0 Let (wX,T ) and (wY,T ) be two m-topological spaces. Then f :(wX,T ) → (wY,T ) is m-continuous iﬀ f : ([X]w,T r) → ([Y ]w, (T 0)r) is an mgp-map for each r ≤ w, where T r, (T 0)r are determined as in Proposition 3.19.

0 w Proof. First let f :(wX,T ) → (wY,T ) be m-continuous and N ∈ [Y ] . Then we have the following three possibilities:

(i) N = 0Y or N = wY ;

(ii) N ∈ T 0;

(iii) N 6∈ T 0.

−1 −1 In the case (i), f (0Y ) = 0X and f (wY ) = wX. Hence,

r −1 r 0 r T (f (0Y )) = T (0X) = w ≥ (T ) (0Y ) and r −1 r 0 r T (f (wY )) = T (wX) = w ≥ (T ) (wY ). 0 0 r 0 In the case (ii), N ∈ T ⇒ (T ) (N) = r. Since f :(wX,T ) → (wY,T ) be m-continuous, it implies that f −1(N) ∈ T and hence T r(f −1(N)) = r. So, T r(f −1(N)) ≥ (T 0)r(N). In the case (iii), N ∈ T 0 → (T 0)r(N) = 0 and hence T r(f −1(N)) ≥ 0 = (T 0)r(N). Therefore, f : ([X]w,T r) → ([Y ]w, (T 0)r) is an mgp-map for each r ≤ w. Converse part follows from Proposition 3.19 and Proposition 3.35.

Proposition 3.37 Let ([X]w, τ), ([Y ]w, τ 0) and ([Z]w, τ 00) be m2-topological spaces, where τ, τ 0 and τ 00 are m-gradtions of openness on [X]w, [Y ]w and [Z]w, respectively. If f : ([X]w, τ) → ([Y ]w, τ 0) and g : ([Y ]w, τ 0) → ([Z]w, τ 00) are mgp-maps, then g ◦ f : ([X]w, τ) → ([Z]w, τ 00) is an mgp-map.

Proof. Proof is straightforward.

Definition 3.38 Let ([X]w, τ) and ([Y ]w, ν) be two m2-topological spaces where τ and ν are counts of openness on [X]w and [Y ]w, respectively. Let M ∈ [X]w, N ∈ [Y ]w and τ M , νN be m-subspace gradations of openness on M and N, respectively. Then f :(M, τ M ) → (N, νN ) is said to be an mgp-map if

τ M (f −1(P ) ∩ M) ≥ νN (P ) for any P ⊆ N.

Proposition 3.39 Let ([X]w, τ) and ([Y ]w, ν) be two m2-topological spaces, where τ and ν are counts of openness on [X]w and [Y ]w, respectively. Let M ∈ [X]w, N ∈ [Y ]w and τ M , νN be m-subspace gradations of openness on M and N, respectively. If f : ([X]w, τ) → ([Y ]w, ν) is an mgp-map and f(M) ⊆ N, then f :(M, τ M ) → (N, νN ) is an mgp- map. On type-2 m-topological spaces [91]

Proof. Let P ⊆ N and A be any member of [Y ]w such that P = A ∩ N. Since f : ([X]w, τ) → ([Y ]w, ν) is an mgp-map, it follows that

τ(f −1(A)) ≥ ν(A). (3)

Now f −1(P ) = f −1(A ∩ N) = f −1(A) ∩ f −1(N). Thus, f −1(P ) ∩ M = f −1(A) ∩ f −1(N) ∩ M = f −1(A) ∩ M, since M ⊆ f −1(f(M)) ⊆ f −1(N). So, by (3), τ M (f −1(P ) ∩ M) = τ M (f −1(A) ∩ M) ≥ τ(f −1(A)) ≥ ν(A). Hence,

τ M (f −1(P ) ∩ M) ≥ ∨{ν(A): A ∩ N = P } = νN (P ).

Therefore f :(M, τ M ) → (N, νN ) is an mgp-map.

Proposition 3.40 Let ([X]w, τ), ([Y ]w, ν) and ([Z]w, ω) be three m2-topological spaces where τ, ν and ω are counts of openness on [X]w, [Y ]w and [Z]w, respectively. Let M ∈ [X]w, N ∈ [Y ]w, P ∈ [Z]w and τ M , νN , ωP be m-subspace gradations of openness on M, N and P , respectively. If f :(M, τ M ) → (N, νN ) and g :(N, νN ) → (P, ωP ) are mgp-maps and f(M) ⊆ N, then the composition mapping g ◦ f :(M, τ M ) → (P, ωP ) is an mgp-map. Proof. Let A ⊆ P . Since g is an mgp-map, it follows that

νN (g−1(A) ∩ N) ≥ ωP (A). (4)

Again since [g−1(A) ∩ N] ⊆ N, f is an mgp-map, we have

τ M M ∩ f −1(g−1(A) ∩ N)) ≥ νN (g−1(A) ∩ N).

This in view of (4) gives

τ M (M ∩ f −1(g−1(A)) ∩ f −1(N)) ≥ ωP (A) and, since M ⊆ f −1(N),

τ M (M ∩ (g ◦ f)−1(A)) ≥ ωP (A).

Therefore, g ◦ f is an mgp-map.

4. Conclusion and future work

In this paper, the concepts of a count of openness, a subspace count of openness, an mgp-maps are introduced. We deﬁne a generalized m-topological space which is called m2-topological space. We have shown that such a count is generated by a descending family of m-topologies and vice versa. The behaviour of the functional image and the functional preimage of an m2-topology, the continuity of the identity mapping and a constant mapping in m2-topologies are also examined. The concepts of topological structures and their generalizations are one of the most powerful notions in branches of science and information systems. It is the generalized methods for measuring the similarity and dissimilarity between the objects [92] Sk. Nazmul in msets as universe. In this sense, this work has a great importance. There is a wide scope for further research to extend it in topological groups theory, which has many applications in abstract integration theory viz. Haar measure, Haar integral etc. and also in manifolds theory through the development of Lie groups.

Acknowledgement. The author express his sincere thanks to the reviewers for their valuable suggestions. The authors are also thankful to the editors-in-chief and managing editors for their important comments which helped to improve the presentation of the paper.

References

[1] Alkhazaleh, Shawkat, and Abdul Razak Salleh, and Nasruddin Hassan. "Soft multisets theory." Appl. Math. Sci. (Ruse) 5, no. 69-72 (2011): 3561–3573. Cited on 77. [2] Babitha, K.V., and Sunil Jacob John. "On soft multi sets." Ann. Fuzzy Math. Inform. 5, no. 1 (2013): 35–44. Cited on 77. [3] Blizard, Wayne D. "Multiset theory." Notre Dame J. Formal Logic 30, no. 1 (1989): 36–66. Cited on 77. [4] de Bruijn, N.G. "Denumerations of rooted trees and multisets." Discrete Appl. Math. 6, no. 1 (1983): 25–33. Cited on 77. [5] Chattopadhyay, Kshitish Chandra and R.N. Hazra, and S.K. Samant. "Gradation of openness: fuzzy topology." Fuzzy Sets and Systems 49, no. 2 (1992): 237–242. Cited on 78. [6] Girish, K.P., and Sunil Jacob John. "Relations and functions in multiset context." Inform. Sci. 179, no. 6 (2009): 758–768. Cited on 77. [7] Girish, K.P., and Sunil Jacob John. "General relations between partially ordered multisets and their chains and antichains." Math. Commun. 14, no. 2 (2009): 193–205. Cited on 77, 78 and 79. [8] Girish, K.P., and Sunil Jacob John. "Multiset topologies induced by multiset relations." Inform. Sci. 188 (2012): 298–313. Cited on 77, 78 and 80. [9] Girish, K.P., and Sunil Jacob John. "On Multiset Topologies." Theory and Appli- cations of Mathematics & Computer Sciences 2, no. 1 (2012): 37–52. Cited on 77, 78 and 80. [10] Hallez, Axel, and Antoon Bronselaer, and Guy De Tré. "Comparison of sets and multisets." Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 17, August 2009, suppl. (2009): 153–172. Cited on 77. [11] Hazra, R.N., and S.K. Samanta, and K.C. Chattopadhyay. "Fuzzy topology rede- ﬁned." Fuzzy Sets and Systems 45, no. 1 (1992): 79–82. Cited on 78. [12] Herawan, T., and M.M. Deris. "On Multi-soft Sets Construction in Information Systems." In: Emerging Intelligent Computing Technology and Applications. With Aspects of Artiﬁcial Intelligence, edited by DS. Huang et all. 101–110. ICIC 2009. Vol. 5755 of Lecture Notes in Computer Science. Berlin, Heidelberg: Springer, 2009. Cited on 77. [13] Hickman, John L. "A note on the concept of multiset." Bull. Austral. Math. Soc. 22, no. 2 (1980): 211–217. Cited on 77. On type-2 m-topological spaces [93]

[14] Höhle, Ulrich, and Alexander P. Šostak. "Axiomatic foundations of ﬁxed-basis fuzzy topology." In: Mathematics of fuzzy sets, 123–272, Handb. Fuzzy Sets Ser., 3, Boston, MA: Kluwer Acad. Publ., 1999. Cited on 78. [15] Kubiak, T. On fuzzy topologies. Ph.D. Thesis. Poznan: A. Mickiewicz University, 1985. Cited on 78. [16] Majumdar, Pinaki. "Soft multisets." J. Math. Comput. Sci. 2, no. 6 (2012): 1700– 1711. Cited on 77. [17] Miyamoto, Sadaaki. "Two generalizations of multisets." In: Rough set theory and granular computing, 59–68. Vol 125 of Stud. Fuzziness Soft Comput., 125. Berlin: Springer, 2003. Cited on 77. [18] Nazmul, Sk., and Pinaki Majumdar, and Syamal K. Samanta. "On multisets and multigroups." Ann. Fuzzy Math. Inform. 6, no. 3 (2013): 643–656. Cited on 78, 79 and 80. [19] Nazmul, Sk., and Syamal K. Samanta. "On soft multigroups." Ann. Fuzzy Math. Inform. 10, no. 2 (2015) 271–285. Cited on 78. [20] Osmanoğlu, İsmail, and Deniz Tokat. "Compact soft multi spaces." Eur. J. Pure Appl. Math. 7, no. 1 (2014): 97–108. Cited on 77. [21] Šostak, Alexander P. "Two decades of fuzzy topology: the main ideas, concepts and results." Russian Math. Surveys 44, no. 6 (1989): 125–186 . Cited on 78. [22] Tokat, Deniz and Ismail Osmanoglu. "Connectedness on Soft Multi Topological Spaces." Journal of New Results in Science 2 (2013): 8–18. Cited on 77. [23] Yager, Ronald R. "On the theory of bags. Internat." J. Gen. Systems 13, no. 1 (1987): 23–37. Cited on 77. [24] Ying, Ming Sheng. "A new approach for fuzzy topology. I." Fuzzy Sets and Systems 39, no. 3 (1991): 303–321. Cited on 78.

Department of Mathematics Bankura University Purandarpur Bankura-722155 West Bengal India E-mail: [email protected]

Received: November 13, 2016; ﬁnal version: November 24, 2017; available online: January 2, 2018.

Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 95-115 DOI: 10.1515/aupcsm-2017-0008

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Edward Tutaj LikeNNN’s – a point of view on natural numbers Communicated by Justyna Szpond

Abstract. We define and study some simple structures which we call likens and which are conceptually near to both sets of natural numbers, i.e. N ∗ with addition and N = N \{0} with multiplication. It appears that there are many different likens, which makes it possible to look on usual natural numbers from a more general point of view. In particular, we show that N ∗ and N are related to some functionals on the space of likens. A similar idea is known for a long time as the Beurling generalized numbers. Our approach may be considered as a little more natural and more general, since it admits the finitely generated likens.

1. Introduction

If we say the set of natural numbers we usually mean the set N = {0, 1, 2,...} equipped with the operations of addition "+" and multiplication "·", and we mean the natural order in N which agree with both operations. The addition and multiplication are related by the distributive law: k · (m + n) = k · m + k · n. If we reject the distributive law, we obtain two diﬀerent structures: (N, +) and (N∗, ·) which – as mathematical structures – are ordered semigroups. First of these semigroups (N, +) is a sub-semigroup of the ordered semigroup R+ = [0, +∞). This is not true for N∗ = {1, 2, 3,...} with multiplication. However the map

∗ + ln : N 3 k → ln k ∈ R is a monomorphism and is increasing. Hence the image ln(N∗) is a sub-semigroup of R+. Here lies the essence of the idea of a liken: this is a sub-semigroup of R+ which is additionally ordered "like N". AMS (2010) Subject Classiﬁcation: 11A41. Keywords and phrases: Beurling numbers; distribution of prime numbers. [96] Edward Tutaj

The idea of the study of the structures, which are more general than the multiplicative semigroup of natural numbers, appeared for the first time eighty years ago in the paper of Beurling [1], with the aim to elucidate if, and how far, the Prime Number Theorem, describing the distribution of prime numbers in the set of natural numbers, depends on addition in N. Since then several authors obtained many interesting theorems. An excellent review on these results is presented in the monograph [2]. To understand the definition of a liken, let us recall the definition of Beurling generalized numbers. Consider an increasing (not necessarily strictly) sequence P of real numbers

1 < p1 ≤ p2 ≤ p3 ... such that lim pn = ∞. n→∞ The numbers which can be written as

n = pk1 · pk2 · pkm i1 i2 im are called generalized integers and they form a multiplicative sub-semigroup B = + B(p1, p2,...) of the multiplicative semigroup R . The elements of the sequence P = (p1, p2,...) play the role of generalized prime numbers. The semigroup B is then generated by its subset P and one may then study the distribution of P in B. Clearly, the family of sub-groups B, which one may obtain in this way, is very rich, and hence one may observe different type of distribution of generalized primes in generalized integers. This is in fact the content of [2]. In the present paper, we would like to present a slightly different point of view on the Beurling numbers. First, it seems to be useful to consider the sub- groups generated by a finite number of generators B(p1, p2, . . . , pk). Obviously, the problem of distribution of primes in such situation is not interesting, but the finitely generated sub-groups seem to be natural objects of this theory. Moreover, we think that because of the habits going from the linear algebra, it is most convenient to work with the additive sub-semigroups of the semigroup of positive reals. The aim we would like to reach, perhaps difficult to obtain, is to create a structure on the set of likens. We make in Part I of this paper some steps in this direction, where we define likens as countable semigroups which tend to infinity, and we make some general observations about likens. The main result of this part of the paper is Theorem 16 which says that two likens are isomorphic if and only if their generators are linearly dependent. The second part of this paper is devoted to the study of the sequence of gaps ∞ in likens. Since a liken L = (xn)0 is a strictly increasing sequence tending to infinity, one may consider the sequence of gaps of L defined as the sequence of differences δk = xk+1 − xk. We prove two theorems, Theorem 25 and Theorem 29, which describe some general properties of the sequence of gaps. In particular, Theorem 29 is a characterization of the additive set of natural numbers N among the family of all likens and this theorem may be confirming the usefulness of the space of likens. Indeed, it appears that only in N the sequence of gaps does not tend to 0 (each gap in N equals 1). The second theorem of this type, which will LikeN’s – a point of view on natural numbers [97] appear in the future work, says that, roughly speaking, the liken N∗ is strongly related to the assumption that its sequence of gaps is strictly decreasing.

2. Part I. General properties of likens

2.1. Deﬁnition of a liken In this paper, and in particular in this section, we will use the following notations:

+ R = [0, ∞), + Q = [0, ∞) ∩ Q, N −→ ∞ R = { a = (ai)1 : ai ∈ R}, + −→ (R )N = { a ∈ RN : ai ≥ 0}, N −→ N R0 = { a ∈ R : ∃j : i > j ⇒ ai = 0}, N −→ ∞ Q = { a = (ai)1 : ai ∈ Q}, + −→ (Q )N = { a ∈ QN : ai ≥ 0}, N −→ N Q0 = {m ∈ Q : ∃j : i > j ⇒ ai = 0}, N −→ N N0 = { a ∈ N : ∃j : i > j ⇒ ai = 0}. (1)

We start by formulating the deﬁnition of a liken.

Definition 1 ∞ A liken L is a sequence (xn)0 of real numbers such that:

a) for all n ∈ N we have 0 = x0 ≤ xn < xn+1,

b) for all m, n ∈ N there is k ∈ N such that xn + xm = xk. It follows directly from Deﬁnition 1 that

Proposition 2 In the notations as above

i) L is a semigroup,

ii) limn→∞ xn = +∞. Proof. The property i) follows directly from a) and b). Let us observe that a) implies inequality 0 < x1, hence L is not trivial. We check by induction that for each k ∈ N the number k · x1 ∈ L. In other words, for each k ∈ N there exists ∞ nk ∈ N such that xnk = k·x1. Hence (xn)0 is an increasing sequence which admits ∞ a subsequence (xnk )1 tending to inﬁnity. This ends the proof of Proposition 2.

Clearly, each liken is a countable sub-semigroup of R+, but the inverse is not true. The semigroup Q+ = [0, ∞) ∩ Q is a sub-semigroup of R+, but it is not a liken. Notice that the property ii) from Proposition 2, although formulated for [98] Edward Tutaj

∞ ∞ a sequence (xn)0 , is in fact a property of the set {xn}0 since the existence of the limit and its value are invariant under the permutations of N. Hence equivalently one may say that a liken L is a countable sub-semigroup of R+ which tends to +∞. This is not the case of Q+. It will be convenient to use the notion of locally finite set. We recall that a subset U of a topological space X is locally finite in X if for each point x ∈ X there exists a neighbourhood V of x such that V ∩ U is a finite set. Using this notion, we may say shortly that the likens are locally + ∞ finite sub-semigroups of R . If L = (xn)0 is a sub-semigroup of the subgroup of non-negative reals R+ (clearly with the usual topology), then the sequence yn = exp(xn) tends to infinity and forms a multiplicative sub-semigroup of the ∞ multiplicative semigroup [1, ∞). Conversely, if G = (xn)1 is a strictly increasing sequence, which is a sub-semigroup of the multiplicative semigroup [1, ∞), then the sequence yn = ln(xn) is a liken. As we will see later, there are no essential differences between likens and Beurling numbers. We will formulate all definitions and results for "additive" likens, but sometimes it will be more convenient to use the multiplicative notations. In each liken we have at the moment two structures: an algebraic structure related to the addition and an ordinal structure related to the ordering in R. Hence we must be precise, what we mean saying "isomorphism of likens". We formulate the formal definitions in order to avoid any misunderstanding. Definition 3 Let (G, +) be a semigroup and let L be a liken. We will say that a map ϕ: G → L is a) an algebraic homomorphism, when ϕ(x + y) = ϕ(x) + ϕ(y), b) an algebraic monomorphism, when it is an injective algebraic homomorphism, c) an algebraic isomorphism, when it is a surjective algebraic monomorphism.

In particular, we know now, what it means that two likens L and K are alge- braically isomorphic. It is also clear that each two likens are isomorphic as ordered spaces, since they are isomorphic to the ordered space (N, ≤). Let us mention that the map N 3 n → xn ∈ L is not (in general) a homomorphism of likens and let us mention also that if ϕ: K → L is an ordinal isomorphism, then it is unique. Finally Definition 4 Two likens L and K are isomorphic if the (unique) ordinal isomorphism is also an algebraic homomorphism.

If L0 is a nontrivial subset of a liken L, which is closed with respect to addition, then L0 is a liken too, and it is natural to say that L0 is a sub-liken of the liken L. In semigroups one can consider the so-called cancellation law, i.e. the following property of a semigroup G: for all a ∈ G, b ∈ G, c ∈ G, a + c = b + c ⇒ b = c.

It is well known, that for each semigroup G with the cancellation law there exists a group G (unique up to an isomorphism) such that, roughly speaking, LikeN’s – a point of view on natural numbers [99]

G = G − G. The reason for which we work with semigroups is that passing from G to G, we loose the order which is "like in N". The order will play a fundamental role in our considerations.

2.2. Undecomposable elements An important consequence of the deﬁnition of a liken (more precisely, a consequence of the order type) is the existence of undecomposable elements. We will see that undecomposable elements in likens play an analogous role as prime numbers in N∗.

Definition 5 Let L be a liken and let u ∈ L. We say that u is undecomposable if

u = v + w, v ∈ L 3 w ⇒ v = 0 ∨ w = 0.

Proposition 6 ∞ Each liken L = (xn)0 has at least one undecomposable element.

Proof. We have observed earlier that x1 > 0. We check that u = x1 is undecomposable. Indeed, suppose that x1 = v + w, where w and v are from L and suppose that v > 0 and w > 0. This means in particular, that 0 < v < x1 (and 0 < w < x1). Thus 0 < v < x1 = v + w which is impossible. This ends the proof.

Proposition 7

Let L be a liken, and let PL be the set of indecomposable elements of L. Then each element of x ∈ L can be written in the form

x = m1 · a1 + m2 · a2 + ... + mk · ak, (2) where m1, m2, . . . , mk ∈ N, a1, a2, . . . , ak ∈ PL and k ∈ N.

Proof. Let PL denote the set of all undecomposable elements of the liken L. It follows from Proposition 6 that PL 6= ∅. The set PL may have only one element, as is in the case of the liken N and PL may have inﬁnitely many elements, as in the case of the liken N∗ (more exactly ln(N∗), but we will write N ∗ for ln(N∗)). Let L0 denote the set of all elements of L which can be represented in the form (2). Clearly, L0 is a sub-liken of L. Suppose that L \ L0 6= ∅. Let y be the minimal 0 0 element of L \ L . Since PL ⊂ L then y∈ / PL. Hence y can be written in the form y = v + w where v and w are both non-trivial. This means that v < y and w < y, and in consequence, by deﬁnition of y, v and w are in L0 which is impossible.

Proposition 7 says that the set PL generates the liken L. It is natural to ask now about the uniqueness of the representation (2). This question is a little more complicated, since we work with natural coeﬃcients, and we will return to this question later. [100] Edward Tutaj

2.3. Universal semigroup We will try now to answer the question: how rich the family of all likens is? In other words, we build something which one may call the space of likens. N Let N0 denote, as in (1), the set of all sequences of natural numbers, with almost all terms vanishing, i.e.

N −→ N0 := { n = (n1, n2,...):(nj ∈ N) ∧ (∃i ∈ N : k > i ⇒ nk = 0)}.

N In the set N0 we may consider the operations: + – addition and · – multiplication N by natural numbers, deﬁned as usually. With these operations N0 is an algebraic structure which may be called semimodule or a cone over N. N We set ek = (0, 0,..., 0, 1, 0,...), i.e. ek is an element of N0 . So we have −→ n = (n1, n2,...) = n1 · e1 + n2 · e2 + ....

∞ Using the terminology from linear algebra, we may say that (ek)1 is a basis N N of the cone N0 . This means precisely that each element from N0 can be, in a unique way, written as a linear combination of (ek)k∈N with the coeﬃcient from N N. Clearly, N0 is a semigroup. The structure of a cone over N is not specially important, but we will use this terminology because of the reasons which will become clear in the next section.

N + 2.4. Homomorphisms from N0 to R

+ N + Clearly, R is a cone over N. A map ϕ: N0 → R will be called a homomorphism of semigroups, when

ϕ(n1 · e1 + n2 · e2 + ...) = n1 · ϕ(e1) + n2 · ϕ(e2) + ....

N + It is evident that a homomorphism ϕ: N0 → R cannot be an epimorphism, N N (surjective homomorphism) since N0 is countable (hence ϕ(N0 ) is countable too) and R+ is uncountable. However there exist the monomorphisms (i.e. injective N + homomorphisms) ϕ: N0 → R . We will now give a more detailed description of this situation. The next proposition shows that the notion of a liken is practically equivalent to the notion the Beurling numbers.

Proposition 8 Each function a: N → R+ can be extended in a unique way to a homomorphism N + a˜: N0 → R by linearity. ∞ Proof. Let a = (ai)1 . Setting a˜(ei) = ai, we obtain a necessary formula

a˜(n1 · e1 + n2 · e2 + ...) = n1 · a1 + n2 · a2 + .... (3)

It is not hard to check that a˜ is a homomorphism.

N + Let H(N0 ; R ) (we will write also simply H) denote the set of all homomorphisms of the two considered cones. This set is also a cone with respect to the natural addition and the scalar multiplication. The same is true about the LikeN’s – a point of view on natural numbers [101] product (R+)N. It follows from Proposition 8 that these two structures (i.e. these two cones) are isomorphic and an isomorphism is for example given by

+ N N + (R ) 3 a → a˜ ∈ H(N0 ; R ).

On the other hand, if ϕ ∈ H then a = (ϕ(ei)) determines ϕ, i.e. ϕ =a ˜. Hence N + + N we can identify the set of homomorphism H(N0 ; R ) with the set (R ) . In the last set we have a natural topology, i.e. the product topology which is metrisable. Usually one considers a metric given by

∞ X |ai − bi| d((a ); (b )) = 2−i · . k k 1 + |a − b | i=1 i i

Moreover, it is well known that (R+)N with respect to this metric is a complete metric space (as a countable product of complete metric spaces). So, from the N + topological point of view, the structure of H(N0 ; R ) is relatively simple. N The set of likens will appear to be a subset of H. If ϕ ∈ H then ϕ(N0 ) is a countable sub-semigroup of R+, but is not a liken in general. For example suppose ∞ + that a = (ai)1 ∈ R has a subsequence which has a finite positive limit. Then N ∞ N a˜(N0 ) cannot be a liken since (ai)1 ⊂ a˜(N0 ) and in likens each element has only a finite number of preceding elements. To avoid some technical complications, we will consider separately the case of finitely generated likens.

2.5. Likens with ﬁnite number of generators

Definition 9

We will say that a liken L is ﬁnitely generated, when the set PL is ﬁnite.

Since the set PL is uniquely determined by L, then so is the number card (PL) which we call the dimension of L. When PL = {a1, a2, . . . , ak} then we write, if necessary, L = L(a1, a2, . . . , ak). As we observed in Proposition 7 each element of x ∈ L(a1, a2, . . . , ak) can be written in the form

x = m1 · a1 + m2 · a2 + ... + mk · ak, (4) but unfortunately in likens the representation (4) in general is not unique, as we have in the case of bases of vector spaces.

Example 10 √ √ 2 Let a1 = 1, a2 = 2 and let a3 = 1 + 2 . It is easy to check that a3 ∈/ L(a1, a2) but 2 · a3 = 2 · a1 + a2. In other words, in the liken L(a1, a2, a3) the uniqueness does not hold.

Hence if we wish to have a liken, in which the representation (4) is unique, we must do an additional assumption. Let us denote the vector space R over the ﬁeld Q by (R, Q). We have the obvious fact [102] Edward Tutaj

Proposition 11 If the real numbers a1, a2, . . . , ak are linearly independent in the vector space (R, Q), then the representation (4) is unique. The likens, which have the property described in Proposition 11, will be called in the sequel likens with uniqueness. A little more complicated is the following observation Proposition 12 For any ﬁnite sequence of positive real numbers u1, u2, . . . , uk, the set L of all real numbers x, which can be written in the form

x = m1 · u1 + m2 · u2 + ... + mk · uk, where m1, m2, . . . , mk ∈ N, is a liken. Proof. Clearly, L is a semigroup, then it remains to show that L tends to +∞. Equivalently, we must show that for each A > 0 the set {x ∈ L : x ≤ A} is finite. Let us fix a number A and let α = min{u1, u2, . . . , uk}. Thus α > 0. Let A n = E( α ) + 1, where E(z) is the integral part of the real number z. It is easy to check that the cardinality of the set {x ∈ L : x ≤ A} is less than nk. We will describe a topological character of the space of finitely generated likens with uniqueness. Let Lk denote the set of all likens with uniqueness which have exactly k undecomposable elements. It follows from Propositions 7, 11 and 12 that Lk can be identified with the set Mk of all increasing sequences of positive reals −→ a = (a1, a2, . . . , ak) which are linearly independent in the vector space (R, Q). Let M k denote the cone of all non-negative and non-decreasing sequences from (R+)k. It is easy to observe that M k is a closed sub-cone of (R+)k and M k has non-empty interior. We have the following fact Proposition 13 k k The set M is a Gδ dense subset of M . −→ k Proof. Let m = (m1, m2, . . . , mk) be a point from Z and let −→ + k M−→m = { a ∈ (R ) : m1a1 + m2a2 + ... + mkak = 0}. + k It is clear that M−→m is a closed subset of (R ) and has empty interior. It is also easy to check that [ Mk = Mk \ M−→m. −→ k m∈Z This ends the proof, since Zk is countable.

2.6. Likens with infinite number of generators Now we will give an analogous description of the set of infinitely generated likens. Let L be a liken and let the set PL = {a1, a2,...} be infinite. We assume that PL is linearly independent in the vector space (R, Q). This assumption is sufficient to have the uniqueness of the representation (2). We have observed in

Proposition 2 that in the case when PL is inﬁnite we must have limk→∞ ak = +∞. The converse is also true. Namely, the following proposition holds LikeN’s – a point of view on natural numbers [103]

Proposition 14 −→ ∞ + N Let a = (ai)1 be a sequence from (R ) which is linearly independent in (R, Q) N and tends to inﬁnity. Then a˜(N0 ) is a liken.

N Proof. Since the image a˜(N0 ) does not change after permuting the set PL and since −→ ∞ a = (a1, a2,...) tends to inﬁnity, we may assume that (ak)1 is increasing and that a1 > 0. Let A > 0. We must show that only a ﬁnite number of members of the set N a˜(N0 ) is less than A. Let j ∈ N be a natural number such that k > j ⇒ ak > A. N Let ~m = (m1, m2,...) be a point from N0 such that the following is true:

i) there is k0 > j such that mk0 ≥ 1, or A ii) there is k0 ≤ j such that mk > . 0 a1 If i) then

a˜(~m) = m1a1 + m2a2 + ... + mjaj + ... + mk0 ak0 . . . > mk0 ak0 > A.

If ii) then

A a˜(~m) = m1a1 + ... + mk0 ak0 + ... + mjaj + . . . > mk0 ak0 > a1 · > A. a1

This implies that the only elements of type a˜(~m) which can be less than A are those with mk = 0 for k > j (since mk is a natural number, mk < 1 implies mk = 0) and mk are equi-bounded for k ≤ j. But the number of such ~m is ﬁnite.

Proposition 14 makes it possible to prove for likens with inﬁnitely many generators and with uniqueness, a theorem similar to Proposition 13. More exactly, we have

Proposition 15 Let L∞ denote the set of all likens with inﬁnite number of generators and with ∞ uniqueness. Then L is a Gδ dense subset in an inﬁnite dimensional complete metric space.

We omit the proof of Proposition 15 since it is similar to the proof of Propo- sition 13.

2.7. A theorem on isomorphism of likens

−→ ∞ −→ ∞ Suppose that we have two sequences a = (ak)1 and b = (bk)1 which generate two likens with uniqueness denoted by La and Lb, respectively. We will prove the following

Theorem 16 In the notations as above the likens and are isomorphic, if and only if there La L−→b exists a positive number λ such that −→a = λ · b . [104] Edward Tutaj

Before starting the proof of this theorem, we must do some preparatory observations. −→ −→ Clearly, when a = λ · b , then La is isomorphic to Lb. So we must prove that −→ −→ if La is isomorphic to Lb, then a = λ · b for some λ > 0. Suppose then that ϕ: La → Lb is an isomorphism (algebraic and ordinal) of likens. The same is true −1 for the map ϕ : Lb → La. First we observe that for each k the element ϕ(ak) ∈ Lb −1 −1 is undecomposable in Lb. Indeed, if ϕ(ak) = t + s implies ak = ϕ (t) + ϕ (s) and thus if t and s are both non-trivial, then so are ϕ−1(t) and ϕ−1(s). Next, by induction we check that for each k the equality ϕ(ak) = bk holds. In −→ N consequence for each m = (m1, m2,...) ∈ N0 we have

ϕ(m1 · a1 + m2 · a2 + ...) = m1 · b1 + m2 · b2 + ...

−→ N or equivalently for each m ∈ N0 ,

ϕ(˜a(−→m)) = ˜b(−→m)

−→ N −→ and ﬁnally we may say that for each m ∈ N0 3 n we have

a˜(−→m) ≤ a˜(−→n ) ⇔ ˜b(−→m) ≤ ˜b(−→n ). (5)

In other words, we have just proved that the condition La isomorphic to Lb implies the condition (5). −→ N −→ ∞ Let us observe that each element a ∈ R = { a = (ai)1 : ai ∈ R} deﬁnes a N −→ linear functional on the vector space R0 = { a : ∃j : i > j ⇒ ai = 0} given by the following formula

∞ −→ −→ X a(x) := h a , x i = ai · xi. (6) 1

N The series in (6) is convergent since each vector x ∈ R0 has only a ﬁnite number of non-zero coordinates.

Definition 17 −→ −→ Given a space (set) F ⊂ RN and two linear functionals a and b , we will say that these functionals agree with respect to the order on the space F, when for each x, y ∈ F the following equivalence holds

a(x) ≤ a(y) ⇔ b(x) ≤ b(y).

We will prove the following

18 Theorem −→ −→ N If two functionals a and b agree with respect to the order on N0 then they agree with respect to the order on RN. LikeN’s – a point of view on natural numbers [105] −→ Proof. Let us assume that −→a and b agree with respect to the order on N. −→ N0 −→ + N i) We will prove that a and b agree with respect to the order on (Q )0 . + N Let us assume that x and y are two vectors from (Q )0 . Then there exists a natural number β > 0 such that the vectors x0 = βx and y0 = βy are in N N0 . Hence a(x) ≤ a(y) ⇔ βa(x) ≤ βa(y) ⇔ a(βx) ≤ a(βy) ⇔ a(x0) ≤ a(y0) ⇔ a(x0) ≤ a(y0) ⇔ b(x0) ≤ b(y0) ⇔ βb(x) ≤ βb(y) ⇔ b(x) < b(y).

Thus i) is proved. −→ −→ + N ii) Now we will prove that a and b agree on (R )0 . Indeed, suppose that + N x and y are two vectors from (R )0 . We may assume, without loss of generality, that x and y have both a ﬁnite support bounded by a number + N k ∈ N. Then there exist two sequences of vectors xn and yn from (Q )0 such that xn tends to x and yn tends to y and all elements of these two sequences have the supports also bounded by k. Hence we have a sequence of equivalences a(xn) ≤ a(yn) ⇔ b(xn) ≤ b(yn). Since the supports of all considered vectors are commonly bounded, we may pass to the limit and we obtain a(x) ≤ a(y) ⇔ b(x) ≤ b(y).

N iii) Suppose now that z ∈ R0 is an arbitrary vector. We have the following equivalence a(z) ≤ 0 ⇔ b(z) ≤ 0. Indeed z = z+ − z−, where z+ and z− denote the positive and negative parts of the vector z, respectively. Thus we have

a(z) ≤ 0 ⇔ a(z+ − z−) ≤ 0 ⇔ a(z+) ≤ a(z−) ⇔ b(z+) ≤ b(z−) ⇔ b(z+ − z−) ≤ 0 ≤ b(z) ≤ 0.

This ends the proof of iii) since for each two non-negative vectors x and y we have

a(x) < a(y) ⇔ a(x − y) < 0 ⇔ b(x − y) < 0 ⇔ b(x) < b(y).

It follows from the above considerations that

19 Theorem −→ −→ N If two functionals a and b agree with respect to the order on N0 then they are linearly dependent.

Proof. Applying Theorem 18, it follows from our assumptions that our functionals −→ N −→ agree with respect to the order on R0 . But this means that a and b have equal [106] Edward Tutaj

N kernels. Indeed, suppose that for an x ∈ R0 we have a(x) = 0. Since by Theorem 18, a(x) = a(0) ⇔ b(x) = b(0) then b(x) = 0. It is known that two functionals are linearly dependent if and only if their kernels are equal. Now we are ready to prove Theorem 16. Proof. We have observed above in formula (5) that if and are isomorphic, −→ La Lb then the functionals −→a and b agree with respect to the order and then by Theorem −→ 19, −→a and b are linearly dependent. We will end this section by the following remark. Remark 20 As we have observed above, given a set of generators (finite, or infinite) – say −→ ∞ ∞ a = (ak)1 – the liken La does not depend on the sequence (ak)1 but depends only on the set of its elements. The unique property we need is to be locally finite. Clearly, each finite set is locally finite, and for infinite sets U (subset of U ⊂ R+) we know that U is locally finite if and only if U tends to +∞.

2.8. Diﬀerent counting functions in likens

−→ ∞ −→ k Suppose that a = (ai)i=1 (or a = (ai)i=1) defines a monomorphism. This N ∞ means that a˜(N0 ) is a strictly increasing sequence La = (xn)n=0. As we have ∞ k ∞ observed above (ai)1 (or (ai)1 ) is a subsequence of the sequence (xn)n=0, and an element xn ∈ La is undecomposable if and only if xn =a ˜(ei) for some i ∈ N (i ≤ k). In other words, only those xn are undecomposable (we will also say prime) which are equal to some ai. The fact that La behaves "like N" makes possible to define and study different counting functions, similar to the well known prime counting function in N∗. Definition 21 −→ For a and for La as above and for a real number x ∈ R we set

πL(x) = card {n ∈ N : xn ≤ x}. (7)

Hence the function πL(x) counts the number of those elements of the sequence xn which are less than x. Unfortunately, it is very diﬃcult to write down precisely the formula for the function πL for a given liken L. However, in two important situations we can do it. If L = N, i.e. L has one generator equal 1, then πL(x) = E(x), where E(x) is integral part of x. If L = N∗ then, as we have observed earlier, xn = ln(n). This implies that x πL(x) = E(e ). The next functions we are going to deﬁne, are directly related with the un- ∞ ∞ decomposable elements. Let L = (xn)0 be a liken, and let (ak)1 be a sequence (increasing) of undecomposable elements of L. For x ∈ R we set

πa(x) = max{k : ak ≤ x} LikeN’s – a point of view on natural numbers [107] and ∗ πa(m) = πa(xm). This last function counts the number of generators, as a function of the index n, of the term xn. It may be considered as a generalization of the prime counting function in the case of the classical liken N∗ which is, as we know, asymptotically n equal to ln(n) . The next theorem is rather rough, but gives some qualitative information about the function πL in the case, when L is ﬁnitely generated. Theorem 22 Suppose that 0 < a1 < a2 < . . . < ak is a sequence of real numbers which are linearly independent in the vector space (R, Q). Let L be a liken generated by the k sequence (ai)1 , i.e. L = L(a1, a2, . . . , ak). Then there exist two polynomials, wL and WL(depending on L) of degree k such that

wL(x) ≤ πL(x) ≤ WL(x).

Proof. The isomorphism a˜ given by (3) deﬁning liken L(a1, a2, . . . , ak) is now de- ﬁned on Nk = N × N × ... × N and has the form −→ a˜: N×N×...×N 3 (m1, m2, . . . , mk) → m1 ·a1 +m2 ·a2 +...+mk ·ak =a ˜(m) ∈ R. −→ Since Nk ⊂ Rk then we may consider the linear functional on Rk given by a˜( x ) = Pk −→ i=1 ai · xi. The points m ∈ N × N × ... × N will be called lattice points. When a real number x ∈ [0, ∞) moves, then the hyperplane

−→ k −→ Hx = { x ∈ R :a ˜( x ) = x} moves in Rk and in each position for x > 0 cuts the cone (Rk)+ and forms a pyramid, or – more precisely – a simplex

−→ k + −→ Sx = { x ∈ R ) : 0 ≤ a˜( x ) ≤ x}.

Now we see that the number πL(x) equals to the number of lattice points in Sx. To evaluate from the above the number of lattice points in Sx, we consider a rectangular parallelepiped formed by the edges of Sx "starting" from the origin. If we denote this rectangular parallelepiped by Px, then since Sx ⊂ Px, we conclude that πL(x) is less than the number of lattice points in Px. The length of the edge x of x lying on the axis generated by ei = (0, 0,..., 1, 0,..., 0) equals , hence the S ai number of lattice points in Px is less than the volume of Px. This implies that as WL we can take the polynomial xk WL(x) = . a1 · a2 · ... · ak

To evaluate πL(x) from below, let us observe that there exists a number d > 0 k (depending on a1, a2, . . . , ak) such that [0, d] ⊂ S1. In consequence, the simplex k Sx contains the product [0, d · x] and thus πL(x) is bounded from below by the [108] Edward Tutaj

k number of lattice points in [0, d · x] . This implies that as wL we can take the polynomial k wL(x) = (d · x − 1) .

The polynomial WL bounds the counting function πL from above, but is far from to be the best upper bound, since the rectangular parallelepiped Px is consid- erably bigger than the prism Sx. The right order of magnitude of πL(x) at inﬁnity is near rather to the Lebesgue measure mk(Sx) which equals

1 xk mk(Sx) = · . k! a1 · a2 · ... · ak

The problem of ﬁnding the exact number of lattice points in Sx is a complicated problem from the discrete geometry (counting lattice points) and the so-called Ehrhart polynomials [3]. However, even this rough information, which is given by Theorem 22, allows to the following

Corollary 23 The set of prime numbers in N∗ is inﬁnite.

Proof. Indeed, as we have observed above, the liken N∗ has an exponential counting function which cannot be controlled from above by any polynomial. Hence N∗ cannot be ﬁnitely generated.

3. Part II. Gaps in likens

The word gap is frequently used to name the diﬀerence between two successive elements of a given sequence. In this section we will prove two theorems about the gaps in likens.

3.1. First theorem

∞ ∞ If L = (xn)0 is a liken, then (xn)0 is strictly increasing and then injective. It seems to be interesting to observe that the sequence of gaps between the elements of L in general must not be injective. In particular, this is the case of ﬁnitely generated likens.

Definition 24 ∞ Let L = L(a1, a2, . . . , ad) = (xn)0 be a liken with d generators a1, a2, . . . , ad which are independent in the vector space (R, Q). The sequence of diﬀerences ∞ δ := (δk)k=0, δk = xk+1 − xk will be called the sequence of gaps of the liken L.

Clearly, by deﬁnition of a liken, δk > 0, since L is strictly increasing. LikeN’s – a point of view on natural numbers [109]

Let us ﬁx x > 0 and let M(x) = {xn ∈ L : xn ≤ x}. Hence

M(x) = {x0, x1, x2, . . . , xk(x)}, where x0 < x1 < x2 < . . . < xk(x) and k(x) = max{n : xn ≤ x}. We see that k(x) = πL(x) (7). We set πδ(x) = card (δ({0, 1, 2, . . . , k(x)})). In the notations as above we have Theorem 25 There exists a polynomial Wd such that deg(Wd) < d and such that πδ(x) ≤ Wd(x). Proof. It is clear that Theorem 25 is true in the case, when d = 1 (one can take W1 ≡ 1), hence in the sequel we may assume that d ≥ 2. It follows from our assumptions on L that each element xk of the liken L can be uniquely represented in the form (2),

1 2 d xk = mk · a1 + mk · a2 + ... + mk · ad,

1 2 d where mk, mk, . . . , mk ∈ N. Hence we have

d d d X i X i X i i δk = mk+1 · ai − mk · ai = (mk+1 − mk) · ai. i=1 i=1 i=1

i i i Putting k = mk+1 − mk, we can write

d X i δk = k · ai, i=1

i where all k are integral numbers. i Some of the numbers k are positive, some may be negative. Since the numbers j δk are all positive then for each k at least one of the numbers k is positive. We set Id = I = {1, 2, . . . , d} and we denote

j Ak = {j ∈ Id : k > 0} and 0 Ak = Id \ Ak. j j j j Setting nk = k for j ∈ Ak and nk = −k for j ∈ Id \ Ak we can write

X j X j δk = nk · aj − nk · aj. (8) 0 j∈Ak j∈Ak

0 Let us observe now that the set Ak cannot be empty. Indeed, suppose that for j i i each j we have k > 0, or equivalently, that mk+1 − mk > 0. Hence, in particular, 1 2 1 1 2 2 (d ≥ 2) k > 0 and k > 0. Thus mk+1 ≥ mk + 1 and mk+1 ≥ mk + 1. This implies [110] Edward Tutaj

1 2 X j xk+1 ≥ mk+1 · a1 + mk+1 · a2 + mk+1 · aj j>2 1 2 X j > mk · a1 + mk+1 · a2 + mk+1 · aj j>2 1 2 X j > mk · a1 + mk · a2 + mk+1 · aj j>2 1 2 X j ≥ mk · a1 + mk · a2 + mk · aj j>2

= xk. Setting 1 2 X j z = mk · a1 + mk+1 · a2 + mk+1 · aj j>2 we see that xk+1 > z > xk. Hence xk and xk+1 cannot be two successive elements of the liken L. In other words, this means that the set Ak is never empty and is never all Id, or equivalently 0 < card Ak < d. Let Pd = {A ⊂ Id : A 6= ∅,A 6= Id}. d The family Pd has 2 − 2 elements. It follows from the above that for each k ∈ N there exists a set A ∈ Pd such that we can rewrite (8),

X j X j δk = nk · aj − nk · aj. (9) j∈A j∈A0

Let us ﬁx A ∈ Pd and let δ(A) denote the set of all δk which can be written in the form (9). Finally, for x > 0 we set

δ(A)(x) = {δk : δk ∈ δ(A), k ≤ k(x)}, and πδ(x, A) = card (δ({0, 1, 2, . . . , k(x)}) ∩ δ(A)(x)). Since X πδ(x) = πδ(x, A),

A∈Pd hence to finish the proof of Theorem 25 it is sufficient to show that for each fixed A ∈ Pd there exists a polynomial WA(x) such that deg WA(x) < d and πδ(x, A) ≤ WA(x). Let us fix a set A, let us fix a real x > 0, and let a natural number k be such that δk ∈ δ(A)(x). Then, as we have observed in (9),

X j X j δk = nk · aj − nk · aj = uk − vk, (10) j∈A j∈A0 where X j uk = nk · aj j∈A LikeN’s – a point of view on natural numbers [111] and X j vk = nk · aj. j∈A0 Hence we have

X j X j X j j xk+1 = mk+1 · aj ≥ mk+1 · aj ≥ (mk+1 − mk) · aj j∈Id j∈A j∈A X j = nk · aj = uk. j∈A

0 Thus if xk+1 ≤ x, then uk ≤ x. Using the same argument (replacing A by A ) we check that vk ≤ x. Now let LA denote the liken generated by those aj for which j ∈ A and let ∗ LA0 denote the liken generated by remaining generators. Let L = LA ∪ LA0 . The set L∗ is a subset of the liken L, then this set can be ordered as an increasing ∗ ∞ sequence. Let L = (γp)0 , and γ0 < γ1 < γ2 .... It follows from the above remark that for each δk ∈ δ(A)(x) there exist two indices p ∈ N, q ∈ N such that p > q and by (10), δk = γp − γq. We will prove that p = q + 1. Indeed, suppose that p > q + 1. Hence we have

uk = γp > γp−1 > γq = vk.

Let ck denote the greatest summand of xk and xk+1. Adding ck to both sides of the above inequality we obtain

xk+1 = uk + ck > γp−1 + ck > vk + ck = xk.

Setting y = γp−1 + ck we have xk < y < xk+1 which is impossible since xk and xk+1 are successive in the liken L. Hence we proved that for each δk ∈ δ(A)(x) 0 00 0 00 and for each u ∈ LA and for each v ∈ LA0 3 v such u ≤ x, v ≤ x, v ≤ x we 0 00 0 00 have that, if δk = u − v = u − v then v = v . Thus the set δ(A)(x) has no more elements that the set UA(x) = {z ∈ LA : z ≤ x}.

Since the liken LA is ﬁnitely generated, the number of its elements increases like a polynomial of degree equal to the number of elements of the set A. But the cardinality of A is less than d, hence we can take WA(x) = W because degW = LA card (A). This ends the proof.

Remark 26 Theorem 25 allows us to point out once more the difference between finitely and infinitely generated likens. In the classical liken N ∗ the sequence of gaps is injective, which means that ωL(x) = k(x) = πL(x) (the number of different elements in the sequence of gaps grows at the same rate as the number of the elements of a liken). In consequence, N ∗ cannot be finitely generated. Let us also remark that in N ∗ two successive elements are always relatively prime. As we have observed in Propositions 7 and 8, each element of a liken with uniqueness can be [112] Edward Tutaj identified with the sequence of its coordinates in the expansion (2), then each element x ∈ L is a function on N with the values in N. Hence it clear what we mean by the support of x. Thus "relatively prime" means clearly that for each ∗ k, supp (xk) ∩ supp (xk+1) = ∅. This property of N is never true for finitely generated likens, since in finite dimensional case there exist elements with full support.

3.2. Some consequences of a certain theorem of Dirichlet We will prove some further theorems on gaps in likens for which we will need some consequences of the well known theorem of Dirichlet, see e.g. [4]. Theorem 27 p For each irrational number α there exists infinitely many rational numbers q , p ∈ , q ∈ such that Z N p 1 α − < . q q2 As an easy consequence of Dirichlet’s theorem we obtain Lemma 28 Let {a1, a2, . . . , ak} be a finite sequence of positive real numbers which satisfies the following condition: for each partition of the set {1, 2, . . . , k} onto non-empty subsets K and M we have P i∈K ai P ∈/ Q. j∈M aj Then, for each η > 0 and for each partition (K,M) as above there exist infinitely many pairs (p, q) of natural numbers such that X X q · ai − p · aj < η. i∈K j∈M Proof. Let us fix a partition (K,M) and a positive number η > 0. Next we apply Theorem 27 for P i∈K ai α = P j∈M aj and we obtain P i∈K ai p 1 P − < 2 j∈M aj q q p for infinitely many rational numbers q . This implies that P X X i∈K ai q · ai − p · aj < q2 i∈K j∈M p for infinitely many q . We choose q so large that P ai i∈K < η q2 and this ends the proof. LikeN’s – a point of view on natural numbers [113]

3.3. Second theorem We will prove another theorem on gaps in likens, saying in particular, that the one-dimensional likens, in which the sequence of gaps is constant, constitute a kind of singularity. Namely, we have the following Theorem 29 ∞ Let L = L(a1, a2, . . . , ad) = (xn)0 denote a liken with d generators (d > 1). Let δn = xn+1 − xn be a sequence of gaps of the liken L. Then

lim δn = 0. n→∞ Proof. It is not hard to observe that it suﬃces to verify our theorem only for d = 2, but we must use also Lemma 28. Since δn > 0 then it is suﬃcient to show that

lim sup δn = 0. n→∞ To obtain a contradiction assume that there exist a number α > 0 and a subsequence n1 < n2 < . . . of the sequence of natural numbers such that δnk > α. Then

δnk = xnk+1 − xnk . Let d X x = mj · a . nk nk j j=1

Since the sequence xnk as a subsequence of the liken L tends to infinity, at least one of d sequences mj is a sequence of real numbers tending to infinity. We will nk consider two cases. Case 1. Assume that for each 1 ≤ j ≤ d we have lim mj = ∞. Let us k→∞ nk divide the set {1, 2, . . . , d} into two non-empty and disjoint subsets K and M and fix a number η < α. Now we apply for Lemma 28 K and L. Consider a pair (p, q) such that X X q · ai − p · aj < η. i∈K j∈M Without loss of generality we may assume that X X 0 < q · ai − p · aj < η. i∈K j∈M Now we choose n so large that 1 ≤ j ≤ d there is mj > max(p, q). k nk Consider a point X X x = (mnk + q) · ai + (mnk − p) · aj. i∈K j∈M Since all coefficients at the generators are positive then x belongs to the liken L and there exists s ∈ N such that x = xs. We will prove that

xnk < xs. Indeed, we have X X xs − xnk = q · ai − p · aj > 0. i∈K j∈M [114] Edward Tutaj

In consequence, we have X X α < δnk = xnk+1 − xnk ≤ xs − xnk = q · ai − p · aj < η ≤ α. i∈K j∈M

This contradiction ends the proof of the Case 1. Case 2. Let us suppose that there exists j such that the sequence mj is bounded nk and let K be the set of all such j. More exactly, K := {j : mj < A }. nk j It follows from our assumption that M = {1, 2, . . . , d}\ K 6= ∅. We choose η < α and apply Lemma 28 for K, M and η as above. We choose a pair (p, q) such that we have the inequality X X 0 < q · ai − p · aj < η i∈K j∈M

and we choose n such that mj > p for j ∈ M. k nk Now we set X X x = (mnk + q) · ai + (mnk − p) · aj i∈K j∈M

and we use the same argument as in Case 1. Since all coeﬃcients at generators are positive then x belongs to L and x = xs for some s ∈ N.

We check that xnk < xs. Indeed, we have X X xs − xnk = q · ai − p · aj > 0. i∈K j∈M

In consequence, X X α < δnk = xnk+1 − xnk ≤ xs − xnk = q · ai − p · aj < η ≤ α i∈K j∈M

and this contradiction ends the proof.

Remark 30 Theorem 29 may by interpreted geometrically. As we have proved in Theorem

22 the function πL(x) behaves like a polynomial WL which has the degree equal to the dimension of L. Hence the number of the elements of L in an interval [x, x + 1] behaves like the derivative W 0 of W , hence tends to inﬁnity with x and, L L in consequence, the average gap tends to 0. Theorem 29 says that, in some sense, there are no irregularities in the sequence of gaps.

References

[1] Beurling, Arne. "Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I." Acta Math. 68, no. 1 (1937): 255–291. Cited on 96. LikeN’s – a point of view on natural numbers [115]

[2] Diamond, Harold G., and Wen-Bin Zhang. Beurling generalized numbers. Vol 213 of Mathematical Surveys and Monographs. Providence, RI: American Mathemat- ical Society, 2016. Cited on 96. [3] Ehrhart, Eugène. "Sur les polyèdres rationnels homothétiques à n dimensions." C. R. Acad. Sci. Paris 254 (1962): 616–618. Cited on 108. [4] Rose, Harvey E. A course in number theory. Second edition. New York: The Clarendon Press; Oxford University Press, 1994. Chapter 13. Cited on 112.

State Higher Vocational School in Tarnów ul. Adama Mickiewicza 8 33-100 Tarnów Poland E-mail: [email protected]

Received: November 5, 2017; ﬁnal version: November 29, 2017; available online: January 15, 2018.

Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 117-120 DOI: 10.1515/aupcsm-2017-0009

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Zenon Moszner Translation equation and the Jordan non-measurable continuous functions Communicated by Justyna Szpond

Abstract. A connection between the continuous translation equation and the Jordan non-measurable continuous functions is given.

It is well known that a continuous function is Lebesgue measurable. It is not true for the Jordan measurability (in short: measurability). We give an example of a non-measurable continuous function by the solution of the translation equation.

1. Continuous solutions of translation equation

Every continuous solution of the translation equation

F (F (x, t), s) = F (x, t + s), (1) where F : I × R → I and I is a non-degenerated interval, is of the form

( −1 hn [hn(g(x)) + t], for g(x) ∈ In, t ∈ R, F (x, t) = S (2) g(x), for g(x) ∈ g(I) \ In, t ∈ R, where g : I → I is a continuous idempotent (g ◦ g = g), In ⊂ g(I) for n ∈ N1 ⊂ N are open and disjoint intervals and hn : In → R are homeomorphisms. Indeed, it is proved in the book [3] that every continuous solution F1 of the translation equation for which F1(x, 0) = x is of the form (1) with g(x) = x. Let F be a continuous solution of the translation equation. The function F1 = F |F (I,R)×R

AMS (2010) Subject Classiﬁcation: 39B12, 39B22, 26A99. Keywords and phrases: Translation equation; Jordan non-measurable function. [118] Zenon Moszner is a continuous solution of the translation equation for which F1(x, 0) = x, since if x = F (x1, t1) for some (x1, t1) ∈ I × R, then

F1(x, 0) = F1(F (x1, t1), 0) = F (F (x1, t1), 0) = F (x1, t1) = x.

Moreover, F (x, t) = F (F (x, t), 0) = F1(F (x, 0), t) and F (x, 0) is a continuous idempotent.

2. Main considerations

Definition A function f : I1 → I2, where I1, I2 are the intervals in R, is said to be measurable if the set {x ∈ I1 : f(x) > a} is measurable for every a ∈ R. Let I ⊂ R be a non-degenerated interval, g : I → I a continuous idempotent such that g(I) is a non-degenerated bounded interval. Let C be a set of the Smith- Volterra-Cantor type in g(I), i.e. let C be a non-measurable set obtained in g(I) 1 as a modiﬁcation of the construction of the Cantor set in which 4 is taken in place 1 of 3 ([1] p.191) (the Cantor set is here not good since it is of Jordan measure zero as a closed set of Lebesgue measure zero). Let In be the components of the open set g(I) \ C. Let F be the function given by the formula (2) with these intervals In and arbitrary homeomorphisms hn : In → R. Fix an arbitrary t0 6= 0. We will prove that the functions f(x) = F (x, t0)−g(x) and −f are continuous and at least one of these functions is non-measurable. Indeed, they are continuous since F and g are continuous functions. We have 1) f(x) = 0 for g(x) ∈ C,

2) f(x) 6= 0 for g(x) ∈ In, n ∈ N1 ⊂ N, otherwise we would have g(x) = −1 F (x, t0) = hn [hn(g(x))+t0] and hn(g(x)) = hn(g(x))+t0, a contradiction. Thus, S S S In = {x ∈ In : f(x) > 0} ∪ {x ∈ In : f(x) < 0} = g(I) ∩ {x ∈ I : f(x) > 0} ∪ g(I) ∩ {x ∈ I : f(x) < 0}. S S The set In is non-measurable since In = g(I) \ C, thus at least one of the sets {x ∈ I : f(x) > 0} and {x ∈ I : f(x) < 0} = {x ∈ I : −f(x) > 0} is non-measurable. The proof is completed. The type of monotonicity of homeomorphisms hn decides partly which function: f or −f, is not measurable., e.g. if t0 > 0 and every hn is increasing, then −1 −1 F (x, t0) = hn [hn(g(x)) + t0] > hn [hn(g(x))] = g(x) S for g(x) ∈ In. Thus we have f(x) > 0 for g(x) ∈ In, hence the function f is non-measurable. This type of monotonicity of hn may be of course diﬀerent for diﬀerent n. Let I be the bounded interval and g(x) = x in (2). In this case the function F (x, 0) = x is evidently measurable. Moreover, for every t0 6= 0, the function F (·, t0): I → I is measurable too: for every real number a the set {x ∈ I : F (x, t0) > a} is an interval, as F (·, t0) is onto, continuous and increasing. Translation equation and the Jordan non-measurable continuous functions [119]

Conclusion The diﬀerence of measurable functions (even continuous) may be non-measurable.

It is known that this situation is impossible for the Lebesgue measurable functions. There exists a continuous solution F of (1) such that for all t ∈ R, functions F (·, t) are non-measurable. Indeed, we put h(x) = d(x, C) + 1 for x ∈ [0, 1) and h(x) = x for x ∈ [1, 2], where C is the above set on the interval [0, 1] and d(x, C) is the distance between x and C. This function h is

1) continuous since the function d(x, C) is continuous ([2] p.103), 2) non-measurable since the set {x ∈ [0, 2] : h(x) > 1} = (I\C) ∪ (1, 2] is non-measurable, 3) an idempotent function since it is the identity function on the range of the function h (h([0, 2]) = [1, 2]).

Thus the function F (x, t) = h(x) for (x, t) ∈ [0, 2]×R is the solution of (1) and F (·, t): [0, 2] → [0, 2] is a continuous, non-measurable function for every t ∈ R.

3. Remark

Proposition There exists a solution F of (1) for which F (·, 0) is measurable and F (·, 1) is non-measurable.

Proof. Let g1 : (0, 1]∩Q → (−∞, 0]∩Q, g2 : (0, 1]\Q → (0, +∞)\Q, g3 : (1, 3)∩Q → (0, +∞)∩Q and g4 : (1, 3)\Q → (−∞, 0]\Q be bijections such that g4((1, 2)\Q) ⊂ (−1, 0]. The function g = g1 ∪ g2 ∪ g3 ∪ g4 is a bijection from (0, 3) onto R. This implies that the function F (x, t) = g−1[g(x) + t] is a solution of (1). The function F (x, 0) = x is evidently measurable. We prove that the function F (·, 1) is non-measurable by proving that the set S = {x ∈ (0, 3) : F (x, 1) > 1} is non-measurable. We have

i) (1, 2) ∩ Q ⊂ S since if x ∈ (1, 2) ∩ Q, then g(x) ∈ (0, +∞) ∩ Q, thus g(x) + 1 ∈ (1, +∞) ∩ Q and this yields that F (x, 1) = g−1[g(x) + 1] ∈ (1, 3) ∩ Q, ii) [(1, 2) \ Q] ∩ S = ∅. Indeed, suppose to the contrary that there exists an x0 ∈ (1, 2) \ Q such that F (x0, 1) > 1. We obtain g(x0) = g4(x0) ∈ (−∞, 0] \ Q, thus g(x0) and g(x0) + 1 are irrational numbers. Moreover, g(x0)g4((1, 2) \ Q) ⊂ (−1, 0] hence g(x0) + 1 ∈ (0, 1] and since g(x0) + 1 is an irrational number, we have g(x0) + 1 ∈ (0, 1] \ Q ⊂ (0, +∞) \ Q. From −1 −1 here F (x0, 1) = g [g(x0) + 1] = g2 [g(x0) + 1] ∈ (0, 1] \ Q. We obtain a contradiction since F (x0, 1)>1.

By i) and ii), the set S is non-measurable. [120] Zenon Moszner

The function F from the above proof is evidently discontinuous since, e.g. the set F ((0, 1], 1) is not an interval.

Question Does there exist a continuous solution of (1) which has the property as in the Proposition?

Such a solution, if it exists, must be of the form (2) with N1 6= ∅ and the function g which is not the identity function (see section 2).

References

[1] Carathéodory, Constantin. Vorlesungen über reelle Funktionen. Leipzig, Berlin: Vieweg, Teubner Verlag, 1927. Cited on 118. [2] Kuratowski, Casimir. Topologie. Vol. I. Warszawa: Polskie Towarzystwo Matem- atyczne, 1952. Cited on 119. [3] Sibirski˘ı,Konstantin S. Introduction to topological dynamics. Leiden: Noordhoﬀ International Publishing, 1975. Cited on 117.

Institute of Mathematics Pedagogical University Podchorążych 2 30-084 Kraków Poland E-mail: [email protected]

Received: November 7, 2017; ﬁnal version: January 11, 2018; available online: January 15, 2018. Ann. Univ. Paedagog. Crac. Stud. Math. 16 (2017), 121-144 DOI: 10.1515/aupcsm-2017-0010

FOLIA 206 Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XVI (2017)

Report of Meeting 17th International Conference on Functional Equations and Inequalities, Będlewo, Poland, July 9–15, 2017

The 17th International Conference on Functional Equations and Inequalities (17th ICFEI), dedicated to the memory of Professor Dobiesław Bry- dak, was held at Będlewo (Poland) in the Mathematical Research and Conference Center (MRCC), on July 9–15, 2017. It was organized by the Department of Mathematics of the Pedagogical University of Cracow. The Scientific Committee of the 17th ICFEI consisted of Professors: Nicole Brillouët-Belluot (France), Dobiesław Brydak (Poland) – honorary chairman, Janusz Brzdęk (Poland) – chairman, Jacek Chmieliński (Poland), Krzysztof Ciepliński (Poland), Roman Ger (Poland), Zsolt Páles (Hungary), Dorian Popa (Romania), Ekaterina Shulman (Poland), Henrik Stetkær (Denmark), László Széke- lyhidi (Hungary), Marek Cezary Zdun (Poland). The Organizing Committee consisted of Janusz Brzdęk (chairman), Jacek Chmieliński (vice-chairman), Zbigniew Leśniak (vice-chairman), Eliza Jabłońska (scientific secretary), Paweł Solarz (technical support), Beata Deręgowska, Paweł Pasteczka, Paweł Wójcik. 48 participants came from 15 countries: Austria (3 participants), Denmark (1), Egypt (1), France (1), Germany (1), Hungary (5), India (1), Iran (3), Japan (1), Morocco (1), Poland (24), Portugal (1), Romania (2), United Kingdom (1) and United States (2). The conference was opened on Monday, July 10, by Professor Janusz Brzdęk, the Chairman of the Scientific and Organizing Committees, who welcomed participants on behalf of the Organizing Committee. The opening address was given by Professor Jacek Chmieliński, the Head of the Department of Mathematics of Ped- agogical University of Cracow. The opening ceremony was completed by a talk of Professor Marek Czerni presenting the life and scientific achievements of Professor Dobiesław Brydak, the creator of ICFEI, who passed away on March 21, 2017. [122] Report of Meeting

During 20 scientific sessions, 40 talks were presented; five of them were longer plenary lectures delivered by Professors Jacek Chmieliński, Zbigniew Leśniak, Adam Ostaszewski, Dorian Popa and Ioan Raşa. The talks were devoted mainly to functional equations and inequalities, iteration theory and their applications in various branches of mathematics, as well as some related topics. In particular, the presented talks concerned classical functional equations such as: Cauchy, Jensen, d’Alembert, Gołąb-Schinzel, Baxter, quadratic, exponential, as well as inequalities (e.g., Hlawka’s or Kedlaya’s type). Moreover, properties of orthogonally additive functions, involutions, convex functions or multivalued mappings were discussed. The problem of Hyers-Ulam stability of some functional equations was also discussed. Finally, answers to some problems from previous meetings were given (e.g., of Butler, Derfel, Baron & Ger, Raşa), as well as, during special sessions, some new open problems and remarks were presented. Some social events accompanied the conference: a picnic on Tuesday night, a banquet on Thursday and the piano recital performed by Professors Marek Czerni and László Székelyhidi on Wednesday evening. On Wednesday afternoon participants visited Poznań, walking the old city streets and visiting the historical museum. The Scientific Committee, on its meeting during the conference, accepted the resignation of Professor Krzysztof Ciepliński from the membership in the Com- mittee. Moreover, the Committee entrusted the chairmanship of the Organizing Committee for the next conference to Professor Jacek Chmieliński. The conference was closed on Saturday, July 15, by Professor Janusz Brzdęk. The subsequent 18th ICFEI was announced to be organized in the year 2019.

1. Abstracts of Talks

Marcin Adam Alienation of the quadratic, exponential and d’Alembert equations Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a ﬁeld of characteristic diﬀerent from 2. Inspired by results obtained in [1] and [2], we study the solutions f, g, h: S → K of Pexider type functional equations f(x + y) + f(x + σy) + g(x + y) = 2f(x) + 2f(y) + g(x)g(y), x, y ∈ S, (1) f(x + y) + f(x + σy) + h(x + y) + h(x + σy) (2) = 2f(x) + 2f(y) + 2h(x)h(y), x, y ∈ S, resulting from summing up the generalized version of the quadratic functional equation with the exponential Cauchy equation and the generalized version of the d’Alembert equation side by side, respectively. We show that under some additional assumptions, equations (1) and (2) force f, g, h to solve the quadratic, exponential and d’Alembert functional equations, respectively.

References

[1] P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59 (2000), 255–261. 17th International Conference on Functional Equations and Inequalities [123]

[2] B. Sobek, Alienation of the Jensen, Cauchy and d’Alembert equations, Ann. Math. Sil. 30 (2016), 181–191.

Javid Ali Stablility and data dependence results for Zamfirescu multivalued mappings Approximating fixed points of a nonlinear operator is one the most widely used techniques for solving differential/integral equations. In view of their concrete applications, it is of great interest to know whether these methods are numerically stable or not. In this presentation, we discuss some new stability and data dependence results for the class of multi-valued Zamfirescu operators. Our results generalize and improve several existing results in literature. It is worth mentioning here that our results are new even for single valued mappings.

Anna Bahyrycz On the stability of functional equation by Baak, Boo and Rassias (joint work with Harald Fripertinger and Jens Schwaiger) We consider the functional equation of the following form

d 1 X X 1 X X rf x + rf x − x r j r j j j=1 S∈ d j6∈S j∈S (`) d d − 1 d − 1 X = − + 1 f(x ) ` ` − 1 j j=1 in the class of functions f mapping a normed space X into Banach space Y (both over the ﬁeld K of characteristic 0), r ∈ R \{0} is given, `, d are ﬁxed integers d satisfying the inequality 1 < ` < d/2, and ` denotes the set of all `-subsets of d = {1, . . . , d}. In [1] the authors determined all odd solutions f : X → Y for vector spaces X,Y over R and r ∈ Q \{0}. In [3] Oubbi considered the same equation but for arbitrary real r 6= 0. Generalizing similar results from [1] he additionally investigates certain stability questions for the equation above, but as for that equation itself for odd approximate solutions only. At the 54th International Symposium on the Functional Equation H. Friper- tinger determined the general solution of this equation (the results were obtained jointly with J. Schwaiger). In our talk we present many stability results for the above equation. They based on the fundamental and very general results in [2], where a priory no additional assumption (oddness) is assumed. The results come from a joint work with H. Fripertinger and J. Schwaiger.

References

[1] C. Baak, D.H. Boo, Th.M. Rassias, Generalized additive mapping in Banach modules and isomorphisms between C∗-algebras, J. Math. Anal. Appl. 314(1) (2006), 150–161. [2] A. Bahyrycz, J. Olko, On stability of the general linear equation, Aequationes Math. 89(6) (2015), 1461–1474. [124] Report of Meeting

[3] L. Oubbi, On Ulam stability of a functional equation in Banach modules, Can. Math. Bull. 60(1) (2017), 173–183.

Karol Baron On the set of orthogonally additive functions with orthogonally additive second iterate Let E be a real inner product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all orthogonally additive functions from E into E with the Tychonoﬀ topology.

Nicole Brillouët–Belluot On a generalization of the Baxter functional equation We determine all continuous solutions f : R → R of the functional equation f(af(x)ky + bf(y)`x + cxy) = f(x)f(y) with a, b, c ∈ R and k, ` ∈ N ∪ {0}. This functional equation generalizes the Baxter functional equation

f(f(x)y + f(y)x − xy) = f(x)f(y) and some generalizations of the functional equation of Goł¸ab-Schinzel

f(f(x)ky + f(y)`x) = f(x)f(y).

Janusz Brzdęk A fixed point theorem and Ulam stability in generalized dq-metric spaces + + Let Y be a nonempty set, R0 denote the set of nonnegative reals, and µ: R0 × + + + R0 → R0 . Let ρ: Y × Y → R0 be a dq µ-metric (dislocated quasi µ-metric), i.e. let the following two conditions be fulfilled: (a) if ρ(x, y) = 0 and ρ(y, x) = 0, then y = x, (b) ρ(x, z) ≤ µ(ρ(x, y), ρ(y, z)) for x, y, z ∈ Y . A fixed point theorem for some spaces of functions (with values in Y ) will + + be presented, under the assumptions that Y is ρ-complete and µ: R0 × R0 → + + + + R0 is continuous (with regard to the usual topologies in R0 and R0 × R0 ) and nondecreasing with respect to each variable (i.e. µ(a, b) ≤ µ(a, c) and µ(b, a) ≤ + µ(c, a) for every a, b, c ∈ R0 with b ≤ c). The theorem has been motivated by the notion of Ulam stability and is a natural generalization and extension of the classical Banach Contraction Principle and some other more recent results.

Jacek Chmieliński On a pexiderization of the orthogonality equation and the orthogonality preserving property We consider problems connected with preservation of the inner product or the orthogonality relation by a pair of mappings. Namely, we study the properties:

hf(x)|g(y)i = hx|yi 17th International Conference on Functional Equations and Inequalities [125] and x⊥y =⇒ f(x)⊥g(y) for all x, y from the joint domain of f and g. As an introduction, the case of a single mapping will be shortly reviewed.

References

[1] J. Chmieliński, Orthogonality equation with two unknown functions, Aequationes Math. 90 (2016), 11–23. [2] R. Łukasik, P. Wójcik, Decomposition of two functions in the orthogonality equation, Aequationes Math. 90 (2016), 495–499. [3] J. Chmieliński, R. Łukasik, P. Wójcik, On the stability of the orthogonality equation and orthogonality preserving property with two unknown functions, Banach J. Math. Anal. 10(4) (2016), 828–847. [4] R. Łukasik, A note on the orthogonality equation with two functions, Aequationes Math. 90 (2016), 961–965. [5] M.M. Sadr, Decomposition of functions between Banach spaces in the orthogonality equation, Aequationes Math., to appear.

Jacek Chudziak A characterization of probability distortion functions of the Goldstein-Einhorn type Probability distortion functions play an important role in various models of decision making under risk. In a literature one can ﬁnd some classes of such functions. In particular, Goldstein and Einhorn [1] introduced the following class apγ g (p) = for p ∈ [0, 1], a,γ apγ + (1 − p)γ where a, γ > 0. In the talk we present a characterization of the Goldstein-Einhorn type probability distortion functions.

References

[1] W.M. Goldstein, H.J. Einhorn, Expression theory and the preference reversal phenomenon, Psychological Review 94 (1987), 236–254.

Bruce Ebanks Linked additive functions We discuss some old and new results about functional equations of the form

n X mk jk x fk(x ) = 0 k=1 for nonnegative integers mk, positive integers jk and additive functions fk mapping an integral domain into itself. If there is no “duplication” of terms (that is, if (mk, jk) 6= (mp, jp) for k 6= p), then each fk is the sum of a linear function and [126] Report of Meeting a derivation of some order. We also update a problem posed by Kannappan and Kurepa in 1970 concerning similar equations of a somewhat more general form.

El-Sayed El-Hady On the analytical solutions of some functional equations During the last few decades a certain structure of functional equations see [1] arises from many interesting applications like e.g. fog computing and wireless networks see [2] e.g.. The general form of such structure is given by

A1(x, y)f(x, y) = A2(x, y)f(x, 0) + A3(x, y)f(0, y) + A4(x, y)f(0, 0) + A5(x, y), where Ai(x, y), i = 1,..., 5, are given polynomials in two complex variables x, y. As far as I know there is no exact-from general solution available for such kind of equations. In this talk I will present some investigations of the analytical solutions of such general class of equations using some special cases.

References

[1] E. El-hady, J. Brzdęk, H. Nassar, On the structure and solutions of functional equations arising from queueing models, Aequationes Math. 91(3) (2017), 445–477. [2] F. Guillemin, C. Knessl, J.S.V. Leeuwaarden, Wireless three-hop networks with steal- ing II: exact solutions through boundary value problems, Queueing Systems 74 (2013), 235–272.

Hojjat Farzadfard Precinct theory: a useful tool for iteration theory Let X be a nonempty set, G be a group and φ: X → G be a map. A function f : X → X is said to be in the realm of φ provided that there exists α ∈ G such that φ(f(x)) = αφ(x) for all x ∈ X. We call α the index of f with respect to φ; it is denoted by indφ(f). The set of all functions which are in the realm of φ is called the realm of φ and is denoted by Realm(φ). The above notions were ﬁrst coined by the author in [1] and then developed in [2]. Let X be a nonempty set. A subset F of the set

F(X) := {f : f is a function of X into itself} is called a semi-precinct if it is the realm of a map φ of X into a group G. If φ is surjective, F is called a precinct. In the present work we discuss some recent applications of the above notions in three areas of iteration theory: 1. The Schröder and Abel equations, 2. (regular) iteration groups, 3. regular iterations.

References

[1] H. Farzadfard, B. Khani Robati, The structure of disjoint groups of continuous functions, Abstr. Appl. Anal. 2012, Article ID 790758, 14 pp. [2] H. Farzadfard, Simultaneous Schröder/Abel equations on the topological spaces, J. Dif- ference Equ. Appl. 21 (2015), 1119–1145. 17th International Conference on Functional Equations and Inequalities [127]

Włodzimierz Fechner Systems of functional inequalities for mappings between rings Assume that X and Y are compact Hausdorﬀ spaces, C(X) and C(Y ) are algebras of all real-valued continuous functions on X and Y , respectively, with pointwise algebraic operations and pointwise order and T : C(X) → C(Y ) is an arbitrary mapping. During the talk we will discuss a few systems of functional inequalities for T . We are especially interested in systems involving Hlawka’s functional inequality. In particular, we will study the following system: ( T (x + y) + T (x + z) + T (y + z) ≥ T (x + y + z) + T (x) + T (y) + T (z), T (x · y) ≥ T (x) · T (y), postulated for all x, y ∈ C(X).

Żywilla Fechner A functional equation motivated by some trigonometric identities (joint work with Włodzimierz Fechner) We deal with the following functional equation

f(xy) + λf(x)f(y) = φ(x, y), where λ is a complex number and f and φ are complex mappings deﬁned on a semigroup. Moreover, we assume an addition formula of trigonometric type for φ. Our research is motivated by some earlier results related to a problem posed by S. Butler in 2003. We discuss some possible questions for future research.

References

[1] S. Butler, Problem no. 11030, Amer. Math. Monthly 110 (2003), 637–639. [2] S. Butler, B.R. Ebanks A Functional Equation: 11030, Amer. Math. Monthly 112 (2005), 371–372. [3] W. Fechner, Ż. Fechner. A functional equation motivated by some trigonometric identities, J. Math. Anal. Appl. 449(2) (2017), 1160–1171.

László Horváth Delay differential and Halanay type inequalities In the present talk we develop a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. The emphasis is put on the rate of convergence based on the theory of the generalized characteristic inequality and equation. The applicability and the sharpness of the results are illustrated by examples.

Eliza Jabłońska Solution of a problem posed by K. Baron and R. Ger (joint work with Taras Banakh) We introduce a new family of ‘small’ sets which is tightly connected with two well known σ-ideals: of Haar-null sets and of Haar-meager sets. We deﬁne a subset [128] Report of Meeting

A of a topological group X to be null-finite if there exists a null-sequence (xn)n∈N in X such that for every x ∈ X the set {n ∈ N : xn + x ∈ A} is finite. Applying null-finite sets we prove that a mid-point convex function f : G → R defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a subset which is not null-finite and whose closure is contained in G. This gives an alternative short proof of a known generalization of Bernstein- Doetsch theorem. Since Borel null-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function f : G → R defined on an open convex subset G of a complete linear metric space X is continuous if it is upper bounded on a Borel subset B ⊂ G which is not Haar-null or not Haar-meager in X. The last result resolves an old problem in the theory of functional equations and inequalities posed by K. Baron and R. Ger in 1983.

References

[1] T. Banakh, E. Jabłońska, Null-ﬁnite sets in metric groups and their applications, arXiv:1706.08155v2 [math.GN] 27 Jun 2017.

Tibor Kiss On Jensen-diﬀerences which are quasidiﬀerences (joint work with Zsolt Páles) Let I ⊆ R be a nonempty open subinterval of R. The aim of the talk is to solve the functional equation

x + y f(x) + f(y) f − = G(g(x) − g(y)), (x, y ∈ I), 2 2 where g : I → R and G: g(I) − g(I) → R are diﬀerentiable functions, f : I → R is continuously diﬀerentiable and g0 does not vanish on I.

References

[1] A. Járai, Gy. Maksa, Zs. Páles, On Cauchy-diﬀerences that are also quasisums, Publ. Math. Debrecen, 65/3-4 (2004), 381–398. [2] T. Kiss, Zs. Páles, On a functional equation related to two variable weighted quasi- arithmetic means, 2017 (submitted).

Zbigniew Leśniak On properties of Brouwer flows and Brouwer homeomorphisms We present properties of Brouwer flows, i.e. flows which contain a Brouwer homeomorphism. In particular, we describe a relationship between the equivalence classes of the codivergency relation and the set of regular points. We also show the corresponding results which concern Brouwer homeomorphisms that are not necessarily embeddable in a flow. These results are obtained under assumptions on existence of invariant lines. Such lines play a similar role as trajectories in the case where a Brouwer homeomorphism is embeddable in a flow. 17th International Conference on Functional Equations and Inequalities [129]

Gyula Maksa Results related to the invariance equation K ◦ (M,N) = K (joint work with Zoltán Daróczy) Let I ⊂ R (the reals) be an interval of positive length, K,M,N : I2 → I be functions. In this talk, we discuss the following problems.

• Suppose that K and M are means in the sense that

K(x, y),M(x, y) ∈ [min{x, y}, max{x, y}](x, y ∈ I),

and the invariance equation holds for the functions K,M and N. Find conditions under which the function N will be a mean itself, as well. We present the solution of this problem supposing that K is Matkowski mean, that is, K(x, y) = (f + g)−1(f(x) + g(y)) (x, y ∈ I),

where f, g : I → R are continuous and strictly monotonic functions in the same sense.

• Under given means K and M, what is the solution N of the invariance equation? In special cases, we give the answer to this question, too.

The main motivation of our investigations is the paper [1].

References

[1] P. Kahlig, J. Matkowski, Invariant means related to classical weighted means, Publ. Math. Debrecen, 89/3 (2016), 373–387.

Renata Malejki On the stability of a generalized Fréchet functional equation with respect to hyperplanes in the parameter space (joint work with Janusz Brzdęk and Zbigniew Leśniak) In this paper we consider the functional equation

A1F (x + y + z) + A2F (x) + A3F (y) + A4F (z) (1) = A5F (x + y) + A6F (x + z) + A7F (y + z), where A1,...,A7 ∈ K are constants belonging to K ∈ {R, C} (R and C denote the fields of real and complex numbers, respectively), in the class of functions F : X → Y , where X is a commutative group and Y is a Banach space over the field K. We study the stability of a generalization of the Fréchet functional equation. In the proof of the main result we use a fixed point theorem to get an exact solution of the equation close to a given approximate solution. Our aim is to study the problem of stability of equation (1) for all possible values of coefficients Ai for i ∈ {1,..., 7}. [130] Report of Meeting

Kazimierz Nikodem Functions generating (m, M, Ψ)−Schur-convex sums (joint work with Silvestru Sever Dragomir) The notion of (m, M, Ψ)−Schur-convexity is introduced and functions generating (m, M, Ψ)−Schur-convex sums are investigated. An extension of the Hardy- Littlewood-Pólya majorization theorem is obtained. A counterpart of the result of Ng stating that a function generates (m, M, Ψ)−Schur-convex sums if and only if it is (m, M, ψ)−Wright-convex is proved and a characterization of (m, M, ψ)−Wright-convex functions is given.

References

[1] C.T. Ng, Functions generating Schur-convex sums, In: General inequalities, 5 (Ober- wolfach, 1986), Internat. Schriftenreihe Numer. Math. 80, Birkhäuser, Basel, 1987, 433–438. [2] K. Nikodem, T. Rajba, S. Wąsowicz, Functions generating strongly Schur-convex sums, C. Bandle et al. (eds.), Inequalities and Applications 2010, International Series of Numerical Mathematics 161, 175–182, Springer Basel 2012. [3] S.S. Dragomir, K. Nikodem, Functions generating (m, M, Ψ)−Schur-convex sums, submitted for publication.

Jolanta Olko On a system of two functional inclusions a Given a < 0 < b, b ∈/ Q and polynomials P,Q satisfying some additional conditions, we study the following system of functional inclusions ( F (x + a) ⊂ F (x) + P (x), x ∈ R, F (x + b) ⊂ F (x) + Q(x), where F is a set-valued function deﬁned on the set of reals. Motivated by [1] we obtain a multivalued counterpart to results concerning inequalities related to single-valued polynomials.

References

[1] J. Schwaiger, Remarks on a paper about functional inequalities for polynomials and Bernoulli numbers, Aequationes Math. 78 (2009), 177–183.

Masakazu Onitsuka On the Hyers–Ulam stability of first-order nonhomogeneous linear difference equations Consider the first-order nonhomogeneous linear difference equation

∆hx(t) − ax(t) = f(t) (1) on hZ, where x(t + h) − x(t) ∆ x(t) = and h = {hk| k ∈ } h h Z Z 17th International Conference on Functional Equations and Inequalities [131] for given h > 0; a is a real number with a 6= −1/h; f(t) is a real-valued function on hZ. We call h the “step-size”. Let I be a nonempty open interval of R. We deﬁne T = hZ ∩ I and ( T \{max T }, if the maximum of T exists, T ∗ = T, otherwise.

Throughout this talk, we assume that T and T ∗ are nonempty sets of R. Note here ∗ that if a function x(t) exists on T , then ∆hx(t) exists on T . Now we will define the Hyers–Ulam stability for Eq. (1). We say that Eq. (1) has the “Hyers–Ulam stability” on T if there exists a constant K > 0 with the following property: Let ε > 0 be a given arbitrary constant. If a function φ: T → R satisfies |∆hφ(t) − aφ(t) − f(t)| ≤ ε for all t ∈ T ∗, then there exists a solution x: T → R of Eq. (1) such that |φ(t) − x(t)| ≤ Kε for all t ∈ T . We call such K a “HUS constant” for Eq. (1) on T . It is easy to see that if a = 0 or a = −2/h, then Eq. (1) does not have the Hyers–Ulam stability on hZ. In this study, we treat only the case of a 6= 0 and a 6= −2/h. Eq. (1) is an approximation of the ordinary differential equation

x0 − ax = f(t), (2) where a is a non-zero real number; f(t) is a continuous real-valued function on R. It is well-known that Eq. (2) has the Hyers–Ulam stability with a HUS constant 1/|a| on R. Furthermore, the solution x(t) of (2) satisfying |φ(t) − x(t)| ≤ ε/|a| for all t ∈ R is the only one (unique), where φ(t) is a continuously differentiable function satisfying |φ0(t) − aφ(t)| ≤ ε for all t ∈ R. That is, for Eq. (2), the minimums of HUS constants on R is 1/|a|. See [2, 4] and the references therein. On the other hand, the following result is obtained by using a result presented by Brzdęk, Popa and Xu [1]. In the case where a 6= 0, −1, −2, f(t) ≡ 0 and h = 1, Eq. (1) has the Hyers–Ulam stability with a HUS constant 1/||a + 1| − 1| on Z. Furthermore, the solution x(t) of (1) satisfying |φ(t) − x(t)| ≤ ε/||a + 1| − 1| for all t ∈ Z is the only one, where φ(t) is a function satisfying |∆1φ(t) − aφ(t)| ≤ ε for all t ∈ Z. Comparing the above results under the assumption f(t) ≡ 0, we see that if a 6= 0 and a > −1, then the minimums of HUS constants of (1) with h = 1 and (2) are the same. However, if a 6= −2 and a < −1, then they are different. Now, an important question arises: How does the step-size influence the minimum of HUS constants for Eq. (1) on hZ? The main purpose of this study is to answer the question.

References

[1] J. Brzdęk, D. Popa, B. Xu, Remarks on stability of linear recurrence of higher order, Appl. Math. Lett. 23(12) (2010), 1459–1463. [2] S.M. Jung, Hyers–Ulam stability of linear differential equations of first order. II, Appl. Math. Lett. 19(9) (2006), 854–858. [3] M. Onitsuka, Influence of the step-size on the Hyers–Ulam stability of first-order homogeneous linear difference equations, Int. J. Difference Equ., to appear. [132] Report of Meeting

[4] D. Popa, I. Raşa, On the Hyers–Ulam stability of the linear diﬀerential equation, J. Math. Anal. Appl. 381(2) (2011), 530–537.

Adam Ostaszewski Asymptotic group actions: associated functional equations and inequalities The theory of regular variation has two classical settings. The first, due to Karamata in 1930, rests on the Cauchy functional equation for additive functions. The second, due to Bojanić and Karamata in 1963 and de Haan (1970 on), rests on the Gołąb-Schinzel functional equation (or Goldie functional equation, depending on context). One can unify these two cases by suitable use of an algebraic approach due to Popa and Javor. This clarifies the interplay between the two formulations above; it also gives a unified proof to the hardest single result in each theory, that on quantifier weakening. Key here is a use of functional inequalities, which apparently complements the existing literature. A less classical but very useful setting is that of Beurling slow and regular variation. This artificial-looking but in fact very natural setting originated in Beurling’s approach to extending the Wiener Tauberian theory, from convolutions to ‘convolution-like’ settings. The resulting Wiener-Beurling Tauberian theorem is the natural tool for the Tauberian theorem for the Borel summability method (and for Riesz means, and Beurling moving averages). As the above use of the term convolution suggests, this hinges on group structure, and group actions; as mention of Tauberian theory suggests, this is analysis, involving limits. We give a topological treatment involving asymptotic group actions.

Lahbib Oubbi On the Ulam-Hyers stability of a functional equation of C. Baak et al. Let X and Y be linear spaces over the ﬁeld K ∈ {R, C}, and f : X → Y be an odd mapping. For any rational number r 6= 0, C. Baak, D.H. Boo, and Th.M. Rassias [1] introduced the following functional equation

Pd Pd i(j) j=1 xj X j=1(−1) xj rf + rf r r i(j)∈{0,1} d P i(j)=` j=1 d ` `−1 X = (Cd−1 − Cd−1 + 1) f(xj), j=1

d p q! where d and ` are positive integers so that 1 < ` < 2 , and Cq := (q−p)!p! , p, q ∈ N with p ≤ q. The authors solved this equation and, whenever Y is a Banach space, they showed that it is Hyers-Ulam stable for r 6= 2. In this talk, we will solve the same equation and show its Hyers-Ulam stability for arbitrary non zero scalar r ∈ K with Y only a sequentially complete locally pseudo-convex space. To this aim, we will ﬁrst give a theorem of Forti-Brzdęk type in uniformizable spaces. 17th International Conference on Functional Equations and Inequalities [133]

References

[1] C. Baak, D.H. Boo, Th.M. Rassias, Generalized additive mapping in Banach modules and isomorphisms between C∗-algebras, J. Math. Anal. Appl. 314 (2006), 150–161. [2] J. Brzdęk, On a method of proving the Hyers-Ulam stability of functional equations on restricted domains, Aust. J. Math. Anal. Appl. 6(1) (2009), 1–10. [3] G.L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 295 (2004), 127–133.

Zsolt Páles Generalizations of the Bernstein–Doetsch theorem (joint work with Carlos Gonzáles, Attila Gilányi, Kazimierz Nikodem and Gari Roa) Given a convex set D of a linear space X and two set-valued maps A, B :(D − D) → 2Y , we consider the Jensen convexity type

F (x) + F (y) x + y + A(x − y) ⊆ cl F + B(x − y) , x, y ∈ D 2 2 and the Jensen concavity type

x + y F (x) + F (y) F + A(x − y) ⊆ cl + B(x − y) , x, y ∈ D 2 2 inclusions and derive their convexity/concavity consequences. These results in- volve the Takagi and Tabor transforms of the set-valued maps A and B and generalize those results that have been obtained for real and vector valued functions.

References

[1] C. González, K. Nikodem, Zs. Páles, G. Roa. Bernstein–Doetsch type theorems for set-valued maps of strongly and approximately convex and concave type, Publ. Math. Debrecen 84(1-2) (2014), 229–252. [2] A. Gilányi, C. González, K. Nikodem, Zs. Páles. Bernstein–Doetsch type theorems with Tabor type error terms for set-valued maps, Set-Valued Var. Anal. 25 (2017), 441–462.

Paweł Pasteczka Weighted Kedlaya inequality (joint work with Zsolt Páles) In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean M the Kedlaya-type inequality

A(x1,M(x1, x2),...,M(x1, . . . , xn)) ≤ M(x1,A(x1, x2),...,A(x1, . . . , xn)) holds for an arbitrary (xn) (A stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if (xn) is a vec- x1 x2 ··· xn tor with corresponding (non-normalized) weights (µn) and M µ1 µ2 . . . µn [134] Report of Meeting denotes the weighted mean then, under analogous conditions on M, the inequality   x1 x2 x1 x2 ··· xn x1 M ...M A  µ1 µ2 µ1 µ2 . . . µn  µ1 µ2 . . . µn   x1 x2 x1 x2 ··· xn x1 A ...A ≤ M  µ1 µ2 µ1 µ2 . . . µn  µ1 µ2 . . . µn

µk holds for every (xn) and (µn) such that the sequence ( ) is decreasing. µ1+...+µk Dorian Popa Ulam stability of linear operators Let K be one of the ﬁelds R or C and X a linear space over the ﬁeld K. A function ρX : X → [0, +∞] is called a gauge if

(i) ρX (λx) = |λ| · ρX (x) for all x ∈ X, λ ∈ K, λ 6= 0;

(ii) ρX (x) = 0 if and only if x = 0.

Let A, B be linear spaces over K, ρA, ρB gauges on A and B and L: A → B a linear operator. We say that L is Ulam stable with constant K ≥ 0 if for every x ∈ A such that ρB(Lx) ≤ 1 there exists z ∈ Ker L with ρA(z − x) ≤ K. We obtain a characterization of Ulam stability for a linear operator L and a representation of the best Ulam constant for L. As applications we give some results on the stability of linear operators acting on normed spaces and of some diﬀerential and partial diﬀerential operators.

References

[1] J. Brzdęk, D. Popa, I. Raşa, Hyers-Ulam stability with respect to gauges, J. Math. Anal. Appl. 453 (2017), 620–628. [2] D. Popa, I. Raşa, On the best constant in Hyers-Ulam stability of some positive linear operators, J. Math. Anal. Appl. 412 (2014), 103–108. [3] D. Popa, I. Raşa, Hyers-Ulam stability of the linear diﬀerential operator with non- constant coeﬃcients, Appl. Math. Comput. 219 (2012), 1562–1568.

Teresa Rajba Muirhead inequality for convex orders and a problem of I. Raşa (joint work with Andrzej Komisarski) We present ([2]) a new, very short proof of a conjecture by I. Raşa (25-years- old problem), which is an inequality involving basic Bernstein polynomials and convex functions. Problem. Prove or disprove that

n X i + j (b (x)b (x) + b (y)b (y) − 2b (x)b (y))f ≥ 0 (1) n,i n,j n,i n,j n,i n,j 2n i,j=0 for each convex function f ∈ C([0, 1]) and for all x, y ∈ [0, 1]. 17th International Conference on Functional Equations and Inequalities [135]

It was aﬃrmed positively very recently by J. Mrowiec, T. Rajba and S. Wą- sowicz [3] by the use of stochastic convex orderings, as well as by U. Abel [1] who simpliﬁed their proof using elementary methods. Inequality (1) is equivalent to the following stochastic convex ordering relation

1 FX+Y ≤cx 2 (FX1+X2 + FY1+Y2 ), (2) where (X,Y ), (X1,X2) and (Y1,Y2) are the pairs of independent binomially distributed random variables such that X,X1,X2 ∼ B(n, x) and Y,Y1,Y2 ∼ B(n, y). Using the usual stochastic order relation ≤st, we give a useful suﬃcient condition for the veriﬁcation of relation (2), which in the case of binomial distributions is equivalent to the I. Raşa inequality (1).

Theorem. Let X and Y be two independent random variables with finite means, such that X ≤st Y or Y ≤st X. Let (X1,X2) and (Y1,Y2) be two pairs of independent random variables such that X,X1,X2, and Y,Y1,Y2 are identically distributed. Then 1 F ≤ (F + F ). (3) X+Y cx 2 X1+X2 Y1+Y2 The sufficient condition, that appears in the above theorem, is satisfied for random variables X and Y with distributions from some families (among oth- ers) of probability distributions: binomial, Poisson, negative binomial, gamma, exponential, beta and Gaussian distributions. Taking these distributions, as an immediate consequence of the above theorem, we can obtain several I. Raşa type inequalities as generalizations of the I. Raşa inequality (1). Our methods allow us to give some extended versions of stochastic convex order (3) as well as the I. Raşa type inequalities. In particular, we prove the Muirhead type inequality for convex orderings for convolution polynomials of probability distributions.

References

[1] U. Abel, A problem of Raşa on Bernstein polynomials and convex functions, arXiv:1609.00234v1 [math.CA] 27 Aug 2016. [2] A. Komisarski, T. Rajba, Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials, arXiv:1703.10634v1 [math.CA] 30 Mar 2017. [3] J. Mrowiec, T. Rajba, S. Wąsowicz, A solution to the problem of Raşa connected with Bernstein polynomials, arXiv:1604.07381v1 [math.AP] 25 Apr 2016, J. Math. Anal. Appl. 446 (2017), 864–878.

Ioan Raşa Entropies, Heun functions and convexity

For a probability distribution (p0(x), p1(x),...) depending on a real parameter P 2 x, the index of coincidence is deﬁned by S(x) = i pi (x). The Rényi entropy and Tsallis entropy are R(x) = − log S(x), respectively T (x) = 1 − S(x). [136] Report of Meeting

We consider a family of probability distributions for which S(x) satisﬁes a Heun diﬀerential equation. Convexity properties of S(x) are investigated; combined with the Heun property, they lead to bounds of the corresponding entropies. In particular, we show that S(x) associated with the binomial distribution is log-convex; this proves a conjecture formulated in [2]. The proof is based on the Heun property of S(x) (see [2]) and the relation between S(x) and the Legendre polynomials (see [1]). The complete monotonicity of entropies (including Shannon entropy) is also discussed.

References

[1] G. Nikolov, Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Raşa, J. Math. Anal. Appl. 418 (2014), 852–860. [2] I. Raşa, Entropies and Heun functions associated with positive linear operators, Appl. Math. Comput. 268 (2015), 422–431.

Maciej Sablik Characterizing exponential polynomials The authors of [1] considered the following equation

m n N X X X λs(y)x fi(αix + βiy) = uk(y)vk(x) + Ps(x)ws(y)e i=1 k=1 s=1

d and proved that under suitable assumptions, every fi : R → C is an exponential polynomial. Using methods from [2] we show the following.

Theorem. If

M r mr X X X r−j j frji(αrjix + βrjiy) x y r=0 j=0 i=1 n N X X λs(y)x = uk(y)vk(x) + Ps(x)ws(y)e k=1 s=1 where αrji, βrji ∈ R \{0} are given scalars, and frji, uk, vk, ws, λs map R into C while Ps are polynomials, then frji are exponential polynomials. Further generalizations, in particular a group-theoretical case, will be presented.

References

[1] J.M. Almira, E.V. Shulman, On certain generalizations of the Levi-Civita and Wilson functional equations, Aequationes Math. DOI 10.1007/s00010-017-0489-4. [2] M. Sablik, Characterizing polynomial functions, talk at the 17th Katowice-Debrecen Winter Seminar, Zakopane 2017. 17th International Conference on Functional Equations and Inequalities [137]

Farzane Sadeghi Boome Two orthogonalities based on norm inequalities (joint work with Farzad Dadipour) In this talk, we present two orthogonalities in a normed linear space which are based on angular distance inequalities. We also show that one of these orthogonalities is equivalent with the Singer orthogonality and compare these two orthogonalities with each other. Finally some related geometric properties of normed linear spaces are discussed and a characterization of inner product spaces is obtained. Acknowledgement The second author is supported by the National Elite Foundation through a grant to Professor Mehdi Radjabalipour.

References

[1] J. Alonso, H. Martini, S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012), 153–189. [2] C. Alsina, J. Sikorska, M.S. Tomas, Norm derivatives and characterizations of inner product spaces, World Scientific, Singapore, 2010. [3] C.R. Diminnie, E.Z. Andalafte, R.W. Freese, Angles in normed linear spaces and a characterization of real inner product spaces, Math. Nachr. 129 (1986), 197–204. [4] I. Singer, Angles abstraits et fonctions trigonométriques dans les espaces de Banac, Acad. R. P. Romîne. Bul. Sti. Sect. Sti. Mat. Fiz. 9 (1957), 29–42. [5] F. Dadipour, F. Sadeghi, A. Salemi, An Orthogonality in Normed Spaces based on an Angular Distance Inequality, Aequationes Math. 90(2) (2016), 281–297.

Cristina Serpa On systems of iterative functional equations (joint work with Jorge Buescu) We formulate a general theoretical framework for systems of iterative functional equations between general spaces. We give general necessary conditions for the existence of solutions such as compatibility conditions (essential hypotheses to ensure problems are well-defined). For topological spaces we characterize continuity of solutions; for metric spaces we find sufficient conditions for existence and uniqueness. Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may have exotic properties. We provide a general construction for finding solutions, including an explicit formula showing how, in certain cases, a solution can be constructively determined. We show the relevance of this for the representation of real numbers.

References

[1] M. Barnsley, Fractal functions and interpolation, Constr. Approx. 2 (1986), 303–329. [2] R. Girgensohn, Functional equations and nowhere diﬀerentiable functions, Aequa- tiones Math. 46 (1993), 243–56. [3] C. Serpa, J. Buescu, Constructive solutions for systems of iterative functional equations, Constr Approx 45 (2017), 273–299. [138] Report of Meeting

[4] C. Serpa, J. Buescu, Explicitly deﬁned fractal interpolation functions with variable parameters, Chaos Solitons Fractals 75 (2015), 76–83. [5] C. Serpa, J. Buescu, Non-uniqueness and exotic solutions of conjugacy equations, J. Diﬀerence Equ. Appl. 21(12) (2015), 1147–1162.

Ekaterina Shulman On Montel’s type statements for mappings on groups Let G be a topological group, f : G → C a continuous function. We call f a polynomial of degree ≤ n if

∆hn+1 ··· ∆h2 ∆h1 f = 0 for any h1, . . . , hn+1 ∈ G. (1) Furthermore, we call f a semipolynomial of degree ≤ n if n+1 ∆h f = 0 for any h ∈ G. (2) We study the following question: whether it is sufficient to check the conditions (1) and (2) only for increments hi from a topologically generating subset E ⊂ G? For polynomials the question is solved affirmatively. For semipolynomials, we prove that the answer is positive if 1. E consists of compact elements, 2. G is commutative and E is finite. In the second case we show that both conditions are essential.

Peter Stadler Curve shortening by short rulers We look at homomorphisms h:(R, +) → (G, ◦) on a Lie group G h(s + t) = h(s) ◦ h(t), h(0) = e, h(1) = g. The restriction of h to the interval [0, 1] is a geodesic. On Riemannian manifolds geodesics are locally shortest lines. The problem is to construct long geodesics. But any curve connecting starting point and end point can be shortened by using a ruler which allows to construct short geodesics. If starting point and end point coincide, the shortened curve is closed, too.

We look at two possibilities: If we skip the starting point (left picture), we get the Birkhoﬀ curve shortening as described in [1]. Else, we get the short ruler method as proposed by [2]. We investigate the diﬀerences we get as we iterate the two methods. 17th International Conference on Functional Equations and Inequalities [139]

References

[1] G.D. Birkhoff, Dynamical Systems, p.135f. Colloquium Publications Vol. 9. AMS, 1927. [2] R. Liedl, N. Netzer, Group theoretic and differential geometric methods for solving the translation equation, in Proceedings of the European Conference on Iteration Theory (ECIT 87), p. 240–252, Caldes de Malavella, 1987. World Scientific, Singapore, 1989.

Henrik Stetkær The kernel of the second order Cauchy difference Let S be a semigroup, not necessarily commutative. The Cauchy difference of f : S → C is the function (Cf)(x, y) := f(xy) − f(x) − f(y) for x, y ∈ S, and the second order Cauchy difference is the iterated function (C2f)(x, y, z) = C{Cf(x, ·)}(y, z) for x, y, z ∈ S. That C2f = 0 is equivalent to Whitehead’s functional equation

f(xyz) + f(x) + f(y) + f(z) = f(xy) + f(yz) + f(xz), x, y, z ∈ S.

We relate various functional equations by showing that the functions f : S → C satisfying C2f = 0 are those of the form f(x) = J(x) + B(x, x), x ∈ S, where J : S → C is a solution of (a version of) Jensen’s functional equation and B : S × S → C is bi-additive.

References

[1] H. Stetkær, The kernel of the second order Cauchy diﬀerence on semigroups. Aequa- tiones Math. 91(2) (2017), 279–288.

Mariusz Sudzik On a problem of Derfel The talk concerns the functional equation

1 1 f(x) = f(x − 1) + f(−2x). (1) 2 2 Last year, during 21st European Conference on Iteration Theory held in Innsbruck, Professor Gregory Derfel posed the following:

Problem. Is there a non-constant bounded continuous function f : R → R satisfying (1)? I will show a partial solution to the Problem. The talk is divided into three parts. In the ﬁrst part I will prove that every bounded continuous solution of equation (1) which attains its extreme value is constant. It is the main result. Its proof is quite elementary. In the second part of my presentation I will discuss the properties of solutions of the equation (1) which do not attaining their extreme values. In this case the Problem is still open. I will show, for example, the following fact [140] Report of Meeting

Lemma. Let a, b ∈ R and let f : R → R be a bounded continuous solution of equation (1). If inf{f(t): t ∈ R} < f(x) < sup{f(t): t ∈ R} for every x ∈ R, then

(i) inf{f(x): x ∈ R} = inf{f(x): x < a} = inf{f(x): x > b},

(ii) sup{f(x): x ∈ R} = sup{f(x): x < a} = sup{f(x): x > b}. The last part of my talk is connected with solutions of equation (1) in a smaller class of functions. For example, we can see that there are no non-trivial, bounded, continuous, with bounded variation solutions of equation (1).

László Székelyhidi Exponential polynomials on affine groups Exponential polynomials are the basic building blocks of spectral synthesis. Recently it has turned out that the fundamental theorem of L. Schwartz about spectral synthesis on the reals can be generalized to several dimensions via some reasonable modification of the original setting. As affine groups play a basic role in this generalization it seems reasonable to study the class of exponential polynomials on these objects.

Peter Volkmann Comparison theorems for functional, diﬀerential, and integral equations (joint work with Gerd Herzog) An abstract result will be given, which can be used to establish comparison theorems as indicated in the title. In particular, this result can be applied to the papers from 2012 (http://doi.org/ 10.1007/978-3-0348-0249-921) and 2016 (http://doi.org/10.5445/IR/1000061837) in the same way as a theorem from 2002 (http://doi.org/10.5445/IR/492002) could be used to derive simpler comparison theorems.

Alfred Witkowski On involutions preserving convexity (joint work with Janusz Matkowski)

It is known that if f : R+ → R is a convex function, then so is g(x) = xf(1/x) This means that the operator F (f)(x) = xf(1/x) is a convexity preserving invo- lution on the space of all real functions on R+. We give a classiﬁcation of the linear operators of the form T (f)(x) = H(x, f ◦ h(x)) sharing the above properties.

2. Problem and Remarks

1. Problem. Find the general solution of the functional equation

C1(x, y)F (x) + C2(x, y)G(y) = 0, (∗) 17th International Conference on Functional Equations and Inequalities [141]

2 ∀x, y ∈ T := {(x, y): xy = (a1 + a2y + a3xy)(b1 + b2x + b3xy)} ⊂ C , such that:

1. Ci(x, y), i = 1, 2 are given functions, in particular rational analytic inside the unit disk,

2. F,G: D → C, where D is the unit disk,

3. The set T is a torus in C2,

4. ai, bi ∈ (0, 1), i = 1, 2, 3,

5. equation (∗) arose recently from a network model.

El-Sayed El-Hady

2. Problem. Assume that the restriction of a continuous function f : R2 → R to any straight line is an exponential polynomial. Is it true that f is an exponential polynomial in two variables? The question is a simple version of a problem posed by J.M. Almira [1, Open Problem 2].

References

[1] J.M. Almira, On Popoviciu-Ionescu functional equation, Ann. Math. Sil. 30 (2016), 5–15. Ekaterina Shulman

3. Remark. During the 16th International Conference on Functional Equations and Inequalities a talk was given concerning the stability of the equation appearing in the title (see [2]). My question about the general solution of the equation itself was answered later by Janusz Brzdęk; see [1, p.196]. Contrary to some assertions in the literature the general solution is not an arbitrary quadratic function, but of the form x → a(x2) with additive a. p √ Let n ∈ N be ≥ 2. Replacing x2 + y2 by n xn + yn in the equation it can be shown that the general solution is given by f(x) = a(xn) with a additive. This is a generalized homogeneous polynomial of degree n. But not every homogeneous polynomial of degree n is of this form: Let b: R → R be additive. Then g with g(x) := b(x)n is a homogenous polynomial od degree n which is not of the form x → a(xn) with additive a unless b is continuous. Proof. Suppose a(xn) = b(x)n for all x. Then a((λx + µy)n) = (b(λx + µy))n for all real x, y and all rational numbers λ, µ. Thus

n n X n X n λlµn−la(xlyn−l) = λlµn−lb(x)lb(y)n−l. l l l=0 l=0 [142] Report of Meeting

With y = 1 and l = 2 we get a(x2) = b(x)2b(1)n−2. This implies that a is continuous, a(x) = cx for all x and some c. But then |b(x)| = pn |c| |x|, which implies that b also continuous. The situation is diﬀerent for C, since there are many discontinuous automor- phisms of C.

References

[1] J. Olko, M. Piszczek, Report of meeting: 16th international conference on functional equations and inequalities, Będlewo, Poland, May 17–23, 2015, Ann. Univ. Paedagog. Crac., Stud. Math. 14 (2015), 163–202. [2] L. Aiemsomboon, W. Sintunavarat, On a new type of stability of a radical quadratic functional equation using Brzdęk’s ﬁxed point theorem, Acta Math. Hungar. 151(1) (2017), 35–46. Jens Schwaiger

3. List of Participants

1. ADAM Marcin, Silesian University of Technology, Gliwice, Poland, email: [email protected] 2. ALI Javid, Aligarh Muslim University, Aligarh, India, email: [email protected] 3. BAHYRYCZ Anna, AGH University of Science and Technology, Kraków, Poland, email: [email protected] 4. BARON Karol, University of Silesia, Katowice, Poland, email: [email protected] 5. BRILLOUËT-BELLUOT Nicole, Ecole Centrale de Nantes, Nantes, France, email: [email protected] 6. BRZDĘK Janusz, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 7. CHMIELIŃSKI Jacek, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 8. CHUDZIAK Jacek, University of Rzeszów, Rzeszów, Poland, email: [email protected] 9. CZERNI Marek, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 10. DADIPOUR Farzad, Graduate University of Advanced Technology, Kerman, Iran, email: [email protected] 11. DERĘGOWSKA Beata, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 12. EBANKS Bruce, University of Louisville, Louisville, United States, email: [email protected] 13. EL-HADY El-Sayed, Suez Canal University, Ismailia, Egypt, email: [email protected] 14. FARZADFARD Hojjat, Shiraz Branch, Islamic Azad University, Shiraz, Iran, email: [email protected] 17th International Conference on Functional Equations and Inequalities [143]

15. FECHNER Włodzimierz, Łódź University of Technology, Łódź, Poland, email: [email protected] 16. FECHNER Żywilla, University of Silesia, Katowice, Poland, email: [email protected] 17. FÖRG-ROB Wolfgang, University of Innsbruck, Innsbruck, Austria, email: [email protected] 18. HORVÁTH László, University of Pannonia, Veszprém,Hungary, email: [email protected] 19. JABŁOŃSKA Eliza, Rzeszów University of Technology, Rzeszów, Poland, email: [email protected] 20. KISS Tibor, University of Debrecen, Debrecen, Hungary, email: [email protected] 21. KOMINEK Zygfryd, University of Silesia, Katowice, Poland, email: [email protected] 22. LEŚNIAK Zbigniew, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 23. MAKAGON Andrzej, Hampton University, Hampton, United States, email: [email protected] 24. MAKSA Gyula, University of Debrecen, Debrecen, Hungary, email: [email protected] 25. MALEJKI Renata, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 26. NIKODEM Kazimierz, University of Bielsko-Biała, Bielsko-Biała, Poland, email: [email protected] 27. OLKO Jolanta, AGH University of Science and Technology, Kraków, Poland, email: [email protected] 28. ONITSUKA Masakazu, Okayama University of Science, Okayama, Japan, email: [email protected] 29. OSTASZEWSKI Adam, London School of Economics, London, United Kingdom, email: [email protected] 30. OUBBI Lahbib, Mohammed V University in Rabat, Rabat, Morocco, email: [email protected] 31. PÁLES Zsolt, University of Debrecen, Debrecen, Hungary, email: [email protected] 32. PASTECZKA Paweł, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 33. POPA Dorian, Technical University of Cluj-Napoca, Cluj-Napoca, Romania, email: [email protected] 34. RAJBA Teresa, University of Bielsko-Biała, Bielsko-Biała, Poland, email: [email protected] 35. RAŞA Ioan, Technical University of Cluj-Napoca, Cluj-Napoca, Romania, email: [email protected] 36. SABLIK Maciej, University of Silesia, Katowice, Poland, email: [email protected] 37. SADEGHI BOOME Farzane, Shahid Bahonar University of Kerman, Kerman, Iran, email: [email protected] 38. SCHWAIGER Jens, University of Graz, Graz, Austria, email: [email protected] [144] Report of Meeting

39. SERPA Cristina, Centro de Matemática Aplicações Fundamentais e Investigação Op- eracional, CMAF-CIO, Lisbon, Portugal, email: [email protected] 40. SHULMAN Ekaterina, University of Silesia, Katowice, Poland, email: [email protected] 41. SOLARZ Paweł, Pedagogical University of Cracow, Kraków, Poland, email: [email protected] 42. STADLER Peter, University of Innsbruck, Innsbruck, Austria, email: [email protected] 43. STETKAER Henrik, Aarhus University, Aarhus, Denmark, email: [email protected] 44. SUDZIK Mariusz, University of Zielona Góra, Zielona Góra, Poland, email: [email protected] 45. SZÉKELYHIDI László, University of Debrecen, Debrecen, Hungary, email: [email protected] 46. TABOR Józef, University of Rzeszów, Rzeszów, Poland, email: [email protected] 47. VOLKMANN Peter, KIT, Karlsruhe, Germany, email: [email protected] 48. WITKOWSKI Alfred, UTP University of Science and Technology, Bydgoszcz, Poland, email: [email protected]

(compiled by Eliza Jabłońska)

Available online: January 15, 2018.

ISSN 2081-545X

Publisher Wydawnictwo Naukowe UP 30-084 Kraków, ul. Podchorążych 2 tel./fax +48-12-662-63-83 tel. +48-12-662-67-56 e-mail: [email protected] Zapraszamy na stronę internetową http://www.wydawnictwoup.pl druk i oprawa Zespół Poligraficzny WN UP