10th Int. Workshop on IFSs, Banská Bystrica, 29 Sept. 2014 Notes on Intuitionistic Fuzzy Sets ISSN 1310–4926 Vol. 20, 2014, No. 4, 32–40

D-posets and effect algebras

Martina Paulínyová

Department of Mathematics, Faculty of Natural Sciences Matej Bel University Tajovského 40, 974 01 Banská Bystrica, Slovakia

Abstract: In the paper two algebraizations of IF-sets families are considered: D-posets [11] and effect algebras [5]. An elementary proof is presented of the fact that D-posets and effect algebras are isomorphic structures [12, 13]. Moreover a product is defined on effect algebras and it is proved that the corresponding algebraic structure is equivalent with the Kôpka D-poset [15, 16]. Keywords: D-poset, Effect algebra, Multiplicative operation. AMS Classification: 03G12, 03B5D.

1 Introduction

MV-algebras presents a natural algebraization of IF - set families. It was proved in [24–26]. From the point of view the cathegory theory it was studied in [6–8]. From the MV-algebra theory results one can obtain some results for IF - sets. Especially important applications gives probability theory on MV-algebras (for a review see [28, 25]). Of course, there are other algebraizations of IF-families: new algebras [3], D-posets [10–12, 17], effect algebras [5], A-posets [31]. In the paper we show that D-posets and effect algebras are equivalent. Both structures have been studied in many papers, so D-posets (e.g. [7, 8, 30]) as effect algebras (e.g. [4, 24]). Very imporant are some applications in quantum structures [14, 15, 20], mainly from the point of view of probability theory. A special case of D-posets are MV-algebras, very near strutures to IF-sets families. They play similar role in quantum structures like Boolean algebras in the classical physics. For the study of conditional probability the MV-algebras with product have been defined and announced. They have been defined independently by Riecanˇ [23], and Montgana [21]. A review of probability theory in MV-algebras is contained in [28] (see also [25]).

32 Therefore Kôpka introduced a product in D-posets [16, 17]. The basic properties of probabil- ity in such D-posets (called Kôpka’s D-posets) were introduced in [27] and then used in [19, 29]. Hence we see that analogous problems can be studied in effect algebras with product. We will define multiplicative operations in D-posets and effect algebras and show, that these structures are equivalent.

2 D-posets and effect algebras

Definition 1 By a D-poset D is considered an algebraic structure D = (D, ≤, −, 0, 1), where:

1. ≤ is a partial ordering on D with the least element 0 and the greatest element 1,

2. − : D × D → D is a partial binary operation, where b − a is defined if a ≤ b and there holds: b − a ≤ b, (1) b − (b − a) = a, (2) a ≤ b ≤ c ⇒ c − b ≤ c − a, (c − a) − (c − b) = b − a. (3)

Definition 2 An effect algebra E is an algebraic structure E = (E, +, 0, 1), where + : E × E → E is a partial binary operation on E with fixed elements 0 and 1 and satisfying the following properties:

∃ a + b ⇔ ∃ b + a, a + b = b + a, (4)

∃ a + (b + c) ⇔ ∃ (a + b) + c, a + (b + c) = (a + b) + c, (5) ∀a ∃! a0, a + a0 = 1 and 00 = 1, (6) ∃ a + 1 ⇒ a = 0, (7)

Proposition 1 Algebraic structures listed above have different definitions (different binary oper- ations, partial ordering relation in D-posets) but they describe the same mathematical object.

Before we prove this statement we notice the partial ordering axioms.

Definition 3 Let A be a set. The relation ≤ is a partial ordering on A if satisfies the following conditions: a ≤ a, ∀ a ∈ A, a ≤ b, b ≤ a ⇒ a = b, a, b ∈ A, (8) a ≤ b, b ≤ c ⇒ a ≤ c, a, b, c ∈ A. (9)

Proof. Let D be a D-poset. We will show now, that there exists corresponding effect algebra.

33 1. Assume, that there exists an element a + b. It means, that ∃ c ∈ D, c ≥ a, b = c − a, then we define

c = a + b. (10)

According to (1) we have c − a ≤ c ⇒ b ≤ c and according to (2) we have c − (c − a) = a, since c − a = b after the substitution we get a = c − b and so c = b + a.

c = a + b = b + a.

We derived the first axiom of effect algebras.

2. Assume, that ∃ a + (b + c). Then element b + c exists and we can write

∃ d ∈ D; d ≥ b, c = d − b.

Then according to (10) d = b + c. There holds

∃ e ∈ D; e ≥ a, d = e − a.

Again according to (10) e = a + d. We substitute d = b + c and get e = a + (b + c). Let ∃ f ∈ D; f ≥ a, b = f − a, then f = a + b. As we mentioned before e = a + d. We use the axiom (4), which we already proved, and get a + d = d + a so d = e − a. We substitute the element d defined before and apply the axiom (4).

e − a = b + c = c + b ⇒ (e − a) − c = b

Since b = f − a, thus we have (e − a) − c = f − a. We apply the axiom (3)

(e − a) − (e − f) = f − a

Then c = e − f ⇒ e = f + c ⇒ e = (a + b) + c.

e = a + (b + c) = (a + b) + c

Property (5) is now proved.

3. Element 1 is defined as the greatest element of D. Then for all a ∈ D there holds a ≤ 1. Denote a0 = 1 − a. According to (10) we get 1 = a + a0. Consider a, a0 ∈ D; a + a0 = 1, a0 = 1 − a and let ∃ a00 ∈ D; a + a00 = 1. Following the commutative law shown before we have a + a00 = a00 + a = 1. Thus a = 1 − a00. Using the axiom (2) we get: a0 = 1 − (1 − a00) = a00 ⇒ a0 = a00 Let a = 0. As proven a + a0 = 1, then a0 = 1 − a = 1 − 0 = 1.

34 4. Assume, that there exists a + 1 ∈ D. Denote a + 1 = x. We proved before, that the operation + is commutative, thus we have a + 1 = 1 + a = x. It follows x ≥ 1, a = x − 1. According to the definition of D-posets is 1 the greatest element, therefore ∀x ∈ D, x ≤ 1. On the basis of partial ordering axiom (8) we have

x ≤ 1, x ≥ 1 ⇒ x = 1

After the substitution we get

a = x − 1 = 1 − 1 = 0.

We showed, that it is possible to derive a corresponding effect algebra to any given D-poset. Now let E = (E, +, 0, 1) be any given effect algebra. We will show, that it is possible to create an equivalent D-poset. We define a partial binary relation ≤ as follows:

a ≤ c ⇔ ∃ b ∈ E; a + b = c.

We will show now, that the relation ≤ satisfies the partial ordering axioms.

1. According to (6) ∀a ∃! a0; a + a0 = 1. Hence for the element 1 holds ∃ 1 + 10 = 1. Applying the axiom (7) we have 10 = 0. Thus:

1 = 1 + 0 = (a + a0) + 0 = (a0 + a) + 0 = a0 + (a + 0).

But 1 = a + a0 = a0 + a, so

a + 0 = a ⇒ a ≤ a.

2. Let a ≤ b and b ≤ a. Then

a ≤ b ⇒ ∃ x ∈ E; a + x = b,

b ≤ a ⇒ ∃ y ∈ E; b + y = a. Following the axiom (5) there holds (a + x) + y = a + (x + y) = a. Denote x + y = z. Then a + z = a. According to the axiom (6) a + a0 = 1. We substitute and compute:

1 = (a + z) + a0 = (z + a) + a0 = z + (a + a0) = z + 1.

We got the equation z + 1 = 1. It follows, that z = 0 and so x + y = 0. Now we consider an element x0 ∈ E.

x0 = x0 + 0 = x0 + (x + y) = (x0 + x) + y = 1 + y.

Following (7) ∃ 1 + y ⇒ y = 0. Thus we have x + 0 = 0 and then x = 0. Finally we get the following equation a + 0 = b ⇒ a = b.

35 3. Let a ≤ b and b ≤ c. Then

a ≤ b ⇒ ∃ x ∈ E; a + x = b,

b ≤ c ⇒ ∃ y ∈ E; b + y = c. Substituting b into the second equation and applying the axiom (5) we get

(a + x) + y = a + (x + y) = c ⇒ a ≤ c.

We proved, that the binary relation ≤ we defined is a partial ordering on E.

Now we define a partial binary operation − : E × E → E. Let a, c ∈ E, a ≤ c. Then

∃ b ∈ E; a + b = c, we denote b = c − a. (11)

Based on this definition we will show, that we can apply axioms of D-posets.

1. Let a ≤ b.Then there exists c ∈ E; a + c = b. According to (4)

a + c = c + a ⇒ c ≤ b.

Following the operation − defined in (11) there holds c = b − a. Thus we get b − a ≤ b

2. From the former part of this proof we have c = b − a, a + c = b. We apply (4): b = a + c = c + a. Thus we have a = b − c and substituting into c = b − a we get a = b − (b − a).

3. Now assume, that a ≤ b, b ≤ c, according to (9) is a ≤ c. Thus there exist elements x, y, z ∈ E satisfying: a + x = b ⇒ x = b − a, b + y = c ⇒ y = c − b, a + z = c ⇒ z = c − a. Since c = b + y, b = a + x, substituting and applying the equation (5) we get

c = (a + x) + y = a + (x + y).

But we defined c = a + z. Thus we get x + y = z. Following (4) z = x + y = y + x, hence x = z − y, y ≤ z. After substituting into this equations we get the form of axiom (3) of D-posets: b − a = (c − a) − (c − b) a c − b ≤ c − a.

We showed, that it is possible to derive a D-poset from any given effect algebra. And however D-posets and effect algebras have different definitions, they describe the same mathematical objects.

36 3 Multiplicative operations

In this section we will extend D-posets and effect algebras by establishing of multiplicative operations. The multiplicative operation for D-posets is defined as follows.

Definition 4 Let D = (D, ≤, −, 0, 1) be a D-poset. By a term Kôpka’s D-poset will be denoted an algebraic structure D = (D, ≤, −, ∗, 0, 1), such that ∗ : D × D → D is a commutative and associative binary operation satisfying the following properties:

a ∗ 1 = a, ∀a ∈ D, (12)

a ≤ b ⇒ a ∗ c ≤ b ∗ c, a, b, c ∈ D, a − (a ∗ b) ≤ 1 − b, a, b ∈ D, (13)

Definition 5 Effect algebra with the multiplicative operation is an algebraic structure E = (E, +, •, 0, 1), where E = (E, +, 0, 1) is an effect algebra and • : E × E → E is commu- tative and associative binary operation satisfying the following conditions:

a • 1 = a, ∀a ∈ E, (14)

∀a, b ∃ c, d; b0 = c + d, a = a • b + c. (15)

We will prove, that Kôpka’s D-posets and effect algebras with the multiplicative operation are equivalent algebraic structures.

Proof. We will show now, that for any given Kôpka’s D-poset there exists an effect algebra with the multiplicative operation. Let D = (D, ≤, −, ∗, 0, 1) be a Kôpka’s D-poset, we will derive each axiom of effect algebras with the multiplicative operations.

1. Assume that a ∈ D. Multiplying by the greatest element 1 is defined by (12) as a ∗ 1 = a and we denote it a • 1 = a.

2. Consider an element a−(a∗b) ∈ D. According to (13) there holds a−(a∗b) ≤ 1−b = b0. Let a − (a ∗ b) = c, then ∃ d ∈ D; b0 = c + d, a = a • b + c. Notice, that we denoted the multiplicative operation in the last equation as •. It is only a formal change to cor- respond with the definition of multiplicative operation on effect algebras established in Definition (5).

We proved, that for each Kôpka’s D-poset we can find corresponding an effect algebra with the multiplicative operation. Now assume a given effect algebra with the multiplicative operation E = (E, +, •, 0, 1). We will show, that we can derive an corresponding Kôpka’s D-poset.

1. Multiplying by the fixed element 1 is defined (14) as a • 1 = a, where a ∈ E. We denote it a ∗ 1 = a.

2. Let a ≤ b. Then ∃ k ∈ D; k = b − a. In other words b = a + k. Thus we have

b ∗ c = a ∗ c + k ∗ c ⇒ a ∗ c ≤ b ∗ c

37 3. Consider a − (a ∗ b) ∈ E. According to (15) we have a = a • b + c. Then c = a − a • b. Following the axiom (6) we can find an element b0: b + b0 = 1 ⇒ b0 = 1 − b. But according to (15) we have b0 = c + d. Hence

1 − b = c + d ⇒ 1 − b ≥ c.

Since a = a•b+c, then c = a−a•b. Now we can see, that the last property of multiplication on Kôpka’s D-poset is satisfied too.

We proved, that Kôpka’s D-poset and effect algebra with the multiplicative operation defined over the same domain are equivalent.

4 Conclusion

Two algebraic generalizations of IF-sets have been considered: Kôpka D-posets and effect alge- bras with product. It was proved that they are isomorphic.

5 Open problem

In [31] a new algebraic system has been defined and studied: A-poset. It was proved that A- posets and D-posets present isomorphic structures. It would be interesting to define a product on A-posets and to show that the corresponding structures are isomoprhic.

References

[1] Atanassov, K., Intuitionistic Fuzzy Sets, Springer, Heidelberg, 1999.

[2] Atanassov, K., On Intuitionistic Fuzzy Sets Theory, Springer, Berlin, 2012.

[3] Atanassov, K., On a new algebraic object. Third scientific session of the mathematical foundations artificial intelligence seminar, Part 1, Sofia 1990, 1–5.

[4] Foulis, J. D., Algebraic measure theory. Atti Sem. Mat. Fis. Univ. Modena, Vol. 48, 2000, 435–461.

[5] Foulis, D. J., M. K. Bennett, Effect algebras and unsharp quantum logics. Found. Phys., Vol. 24, 1994, 1331–1352.

[6] Fric,ˇ R., M. Papco,ˇ On probability domains. Internat. J. Theoret. Phys., Vol. 49, 2010, 3092–3100.

[7] Fric,ˇ R., M. Papco,ˇ On probability domains II. Internat. J. Theoret. Phys., Vol. 50, 2011, 3778–3786.

[8] Fric,ˇ R., M. Papco,ˇ On probability domains III. Internat. J. Theoret. Phys. (submitted)

38 [9] Chovanec, F., Difference Posets and their Graphical Representation (in Slovak). Liptovský Mikuláš, 2014.

[10] Chovanec, F., E. Drobná, On a construction of linearly ordered totaly nonatomic D-posets. Proc. Third Seminar Fuzzy sets and Quantum Structures, 2003, 12–27.

[11] Chovanec, F., F. Kôpka, D-posets. Math. Slovaca, Vol. 44, 1994, 21–34.

[12] Chovanec, F., F. Kôpka, D-posets. In: Integral, Measure, and Ordering (B. Rieˇcanand T. Neubrunn), Kluwer, 1997, 278–311.

[13] Kôpka, F., F. Chovanec, D-posets. In: Handbook of Quantum Logic and Quantum Structures, Elsevier, Amsterdam, 2007, 367–428.

[14] Dvurecenskijˇ A., M. Kuková, Observables on quantum structures. Information Sciences, 2013.

[15] Dvurecenskijˇ A., S. Pulmannová, New Trends in Quantum Structures, Kluwer, Dordrecht, 2000.

[16] Kôpka, F., D-posets with meet function. Tatra Mt. Math. Publ., Vol. 1, 1992, 83–87.

[17] Kôpka, F., Quasi product on Boolean D-posets. Inter. J. Theor. Phys., Vol. 47, 2008, 26–35.

[18] Kôpka, F., F. Chovanec, D-posets. Math. Slovaca, Vol. 44, 1994, 21–34.

[19] Kuková, M., Strong law of large numbers on the Kopkaˇ D- posets. In: FSTA 2012, 79.

[20] Kuková, M., M. Navara, Central limit theorem and laws of large numbers in quantum structures. In: 11th Bienal IQSA Meeting, Cagliari 2012, 44–45.

[21] Montagna, F., An algebraic approach to propositional fuzzy logic. J. Logic. Lang. Inf., Special Issue on Logics of Uncertainty (Mundici, D. ed.), Vol. 9, 2000, 91–124.

[22] Papco,ˇ M., On effect algebras. Soft Computing, Vol. 12, 2007, 26–35.

[23] Riecan,ˇ B., On the product MV-algebras. Tatra Mt. Math. Publ., Vol. 16, 1999, 299–306.

[24] Riecan,ˇ B., On a new approach to probablity theory on MV algebras, Proc. of Workshop on Generalized Nets, Intutionistic Fuzzy Sets and Knowledge Engeneerng, London, 2010, 57–65.

[25] Riecan,ˇ B., Analysis of Fuzzy Logic Models. In: Intelligent Systems (V. M. Koleshko ed.), INTECH 2012, 219–240.

[26] Riecan,ˇ B., Embedding of IF-states to MV-algebras. Proc. of IWIFSGN, Warsawa, 2014. (in press)

39 [27] Riecan,ˇ B., L. Lašová, On the probability on the K opka D-posets. In: Developments in Fuzzy Sets, Intuitionistic Fuzzy sets, Generalized Nets and Related Topics. Vol. I (K. Atanassov et al. eds.), Warsaw, 2010, 167–176.

[28] Riecan,ˇ B., D. Mundici, Probability on MV-algebras. In: Handbook of Measure Theory, Elsevier, Amsterdam, 2002, 869–910.

[29] Samuelcík,ˇ K., I. Hollá, Conditional probability on the Kopkaˇ D-posets. Acta Mathematica Sinica, Vol. 28, 2012, 2197–2204.

[30] Samuelcík,ˇ K., I. Hollá, Disjointness relations on D-posets. Proc. of 2nd International Work- shop on Generalized Nets, Intuitionistic Fuzzy Sets and Knowledge Engineering, July 2010, London, 71–74.

[31] Skrivánek,ˇ V., R. Fric,ˇ R.: Generalized random events (submitted).

[32] Vrábelová, M., On the conditional probability in product MV-algebras. Soft Computing, Vol. 4, 2000, 58–61.

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