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Sheffer Stroke Bg-Algebras by Filters and Homomorphisms BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Bull. Int. Math. Virtual Inst., Vol. 11(2)(2021), 375-386 DOI: 10.7251/BIMVI2102375O Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) SHEFFER STROKE BG-ALGEBRAS BY FILTERS AND HOMOMORPHISMS Tahsin Oner and Tugce Kalkan Abstract. In this paper, a Sheffer stroke BG-algebra, a subalgebra and a normal subset of a Sheffer stroke BG-algebra are given. After determining a filter of Sheffer stroke BG-algebra, a congruence relation on a Sheffer stroke BG-algebra is defined by means of its filter, and quotient of a Sheffer stroke BG-algebra by a congruence relation on this algebra is built. Besides, a homo- morphism between Sheffer stroke BG-algebras is introduced and it is presented that their features are preserved under this homomorphism. 1. Introduction In 1996, Y. Imai and K. Is´eki introduced a new algebraic structure called BCK- algebra. K. Is´eki introduced the new idea be called BCI-algebra [5]. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In 2002, J. Neggers and H. S. Kim [10] constructed a new algebraic structure. They took some properties from BCI and BCK-algebras be called B-algebra. In 2005, C. B. Kim and H. S. Kim [6] introduced the notion of a BG-algebra which is a generalization of B-algebras. A non-empty set X with a binary operation ∗ and a constant 0 satisfying some axioms will construct an algebraic structure be called BG-algebra. With these ideas, fuzzy subalgebras of BG-algebra were developed by Ahn and Lee [2]. Muthuraj et al. [8] presented fuzzy ideals in BG-algebra in 2010. Muthuraj and Devi [9] introduced the concept of multi-fuzzy subalgebra of BG-algebra in 2016. On the other hand, H. M. Sheffer defined originally Sheffer stroke (or Sheffer operation) in 1913 [17]. Since any Boolean function or operation can be stated by 2010 Mathematics Subject Classification. Primary 06F05; Secondary 03G25. Key words and phrases. BG-algebra, Sheffer stroke, filter, congruence, homomorphism. Communicated by Daniel A. Romano. 375 376 ONER AND KALKAN only this operation [7], it attracts many researchers' attention. It also causes to reduction of axiom systems of many structures. So, many researchers want to use this operation on their studies. Also, some applications of this operation has been appeared in algebraic structures such as Sheffer stroke non-associative MV-algebras [3] and filters [14], (fuzzy) filters of Sheffer stroke BL-algebras [15], Sheffer stroke Hilbert algebras [12] and filters [13], Sheffer stroke UP-algebras [16], Sheffer stroke BG-algebras [11] and Sheffer operation in ortholattices [4]. After giving basic definitions and notions about a Sheffer stroke BG-algebra, a subalgebra and a normal subset of a Sheffer stroke BG-algebra, their features are presented. A filter on a Sheffer stroke BG-algebra is defined. It is shown that the family of all filters of a Sheffer stroke BG-algebra forms a complete lattice, and for a subset of a Sheffer stroke BG-algebra, there exists the minimal filter containing this subset. It is proved that a subalgebra of a Sheffer stroke BG-algebra is a filter. Then a medial Sheffer stroke BG-algebra is determined and it is indicated that every non-empty subset of a medial Sheffer stroke BG-algebra is a filter. Besides, it is proved that the intersection of arbitrary number of filters of a Sheffer stroke BG- algebra is a filter while the union of arbitrary number of filters of a Sheffer stroke BG-algebra is generally not a filter. Morever, defining a homomorphism between Sheffer stroke BG-algebras, it is demonstrated that mentioned notions such as filter is preserved under this homomorphism. A kernel of a homomorphism is built and it is proved that the kernel is a filter. Finally, a congruence relation on a Sheffer stroke BG-algebra is determined by its filter and related concepts are given. It is shown that a quotient of a Sheffer stroke BG-algebra defined by a congruence relation is a Sheffer stroke BG-algebra. It is proved that the kernel is a normal subalgebra. 2. Preliminaries In this part, we give the basic definitions and notions about a Sheffer stroke, a BG-algebra and a Sheffer stroke BG-algebra. Definition 2.1. ([4]) Let A = hA; ji be a groupoid. The operation j is said to be Sheffer stroke if it satisfies the following conditions: (S1) a1ja2 = a2ja1; (S2) (a1ja1)j(a1ja2) = a1; (S3) a1j((a2ja3)j(a2ja3)) = ((a1ja2)j(a1ja2))ja3; (S4) (a1j((a1ja1)j(a2ja2)))j(a1j((a1ja1)j(a2ja2))) = a1: Definition 2.2. [6] A BG-algebra is a non-empty set A with a constant 0 and a binary operation ∗ satisfying the following axioms: (BG:1) a1 ∗ a1 = 0, (BG:2) a1 ∗ 0 = a1, (BG:3) (a1 ∗ a2) ∗ (0 ∗ a2) = a1; for all a1; a2 2 A. A BG-algebra is called bounded if it has the greatest element. SHEFFER STROKE BG-ALGEBRAS BY FILTERS AND HOMOMORPHISMS 377 Definition 2.3. ([6]) A nonempty subset S of a BG-algebra A is called a BG-subalgebra if a1 ∗ a2 2 S, for all a1; a2 2 S. Definition 2.4. ([10]) Let A be a BG-algebra. A nonempty subset N of A is said to be normal if (a1 ∗ x) ∗ (a2 ∗ y) 2 N, for any a1 ∗ a2, x ∗ y 2 N. Definition 2.5. ([1]) A BG-algebra A is called medial if a1 ∗ (a1 ∗ a2) = a2, for all a1; a2 2 A. Definition 2.6. ([11]) A Sheffer stroke BG-algebra is an algebra (A; j; 0) of type (2; 0) such that 0 is the constant in A and the following axioms are satisfied: (sBG:1) (a1j(a1ja1)j(a1j(a1ja1) = 0 , (sBG:2) (0j(a2ja2))j((a1j(a2ja2))j(a1j(a2ja2))) = a1ja1, for all a1; a2 2 A. Lemma 2.1 ([11]). Let A be a Sheffer stroke BG-algebra. Then the following features hold for all a1; a2; a3 2 A: (1) (0j0)j(a1ja1) = a1; (2) (a1j(0j0))j(a1j(0j0)) = a1; (3) (a1j(a2ja2))j(a1j(a2ja2)) = (a3j(a2ja2))j(a3j(a2ja2)) implies a1 = a3, (4) (0j(0j(a1ja1))) = a1ja1; (5) If (a1j(a2ja2))j(a1j(a2ja2)) = 0 then a1 = a2, (6) If (0j(a1ja1)) = (0j(a2ja2)) then a1 = a2, (7) (((a1j(0j(a1ja1)))j(a1j(0j(a1ja1))))j(a1ja1)) = a1ja1, (8) (a1j(a1ja1))j(a1ja1) = a1. Definition 2.7. ([11]) A non-empty subset S of a Sheffer stroke BG-algebra A is called a Sheffer stroke BG-subalgebra of A if (a1j(a2ja2))j(a1j(a2ja2)) 2 S, for all a1; a2 2 S. Definition 2.8. ([11]) Let A be a Sheffer stroke BG-algebra. A non-empty subset N of A is said to be normal subset of A if (((a1j(xjx))j(a1j(xjx)))j(a2j(yjy)))j(((a1j(xjx))j(a1j(xjx)))j(a2j(yjy))) 2 N; for any (a1j(a2ja2))j(a1j(a2ja2)); (xj(yjy))j(xj(yjy)) 2 N. Theorem 2.1 ([11]). Every normal subset N of a Sheffer stroke BG-algebra A is a Sheffer stroke BG-subalgebra of A. Lemma 2.2 ([11]). Let N be a Sheffer stroke normal subalgebra of a Sheffer stroke BG-algebra A and let a1; a2 2 N. If (a1j(a2ja2))j(a1j(a2ja2)) 2 N, then (a2j(a1ja1))j(a2j(a1ja1)) 2 N: 3. On filters of Sheffer stroke BG-algebras We introduce the notion of a filter in a Sheffer stroke BG-algebra in this section. 378 ONER AND KALKAN Definition 3.1. A filter of a Sheffer stroke BG-algebra A is a nonempty subset F ⊆ A which satisfies the following: (SF:1) If a1; a2 2 F , then (a2j(a2j(a1ja1)))j(a2j(a2j(a1ja1))) 2 F and (a1j(a1j(a2ja2)))j(a1j(a1j(a2ja2))) 2 F: (SF:2) If a1 2 F and (a1j(a2ja2))j(a1j(a2ja2)) = 0, then a2 2 F . Theorem 3.1. Let A be a Sheffer stroke BG-algebra. Then the family KA of all filters of A forms a complete lattice. T Proof. f g 2 Let Fi i2I be a family of filters of A. If a1; a2 i2I Fi, then a1; a2 2 Fi, for all i 2 I. Since Fi is a filter of A, then we have (a2j(a2j(a1ja1)))j(a2j(a2j(a1ja1))); (a1j(a1j(a2ja2)))j(a1j(a1j(a2ja2))) 2 Fi: Then \ (a2j(a2j(a1ja1)))j(a2j(a2j(a1ja1))); (a1j(a1j(a2ja2)))j(a1j(a1j(a2ja2))) 2 Fi: 2 T T i I 2 j j j j j 2 (i) Suppose that a1 i2I Fi and (a1 (a2 a2)) (a1 (a2 a2)) i2I Fi holds for 2 2 j j j j j 2 a1; a2 A, that is a1 Fi and (a1 (a2 a2)) (a1 (a2 a2)) = 0 holdT for all i I. Then it is obtained from (SF.2) that a 2 F for all i 2 I. Then a 2 F . 2 i 2 i2IS i T (ii) Let η be the family of all filters ofVA containingT the unionW i2I Fi. ThenT η is a filterV W of A from (i). If we define i2I Fi = i2I Fi and i2I Fi = η, then (KA; ; ) is a complete lattice. Lemma 3.1. Let A be a Sheffer stroke BG-algebra and B be a subset of A. Then there is the minimal filter hBi containing the subset B.
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