On atoms in BCC-algebras

Transcription

1 Discussiones Mathematicae ser. Algebra and Stochastic Methods 15 (1995), On atoms in BCC-algebras Wies law A. DUDEK and Xiaohong ZHANG Abstract We characterize the the set of all atoms of a BCC-algebra and prove that a BCC-algebra which contains only atoms is a BCK-algebra. 1. Introduction. By an algebra G = (G,, 0) we mean a non-empty set G togeter with a binary multiplication and a some distinguished element 0. In the sequel a multiplication will be denoted by juxtaposition. Dots we use only to avoid repetitions of brackets. For example, the formula ((x y) (z y)) (x z) = 0 will be writen as (xy zy) xz = 0. An algebra (G,, 0) is called a BCC-algebra if it satisfies the following axioms: (1) (xy zy) xz = 0, (2) xx = 0, (3) 0x = 0, (4) x0 = x, (5) xy = yx = 0 implies x = y. The above definition is a dual form of the ordinary definition (cf. [1], [5], [6]). In our convention any BCK-algebra is a BCC-algebra, but there are BCC-algebras which are not BCK-algebras (cf. [2]). Such BCCalgebras are called proper. In [2] is proved that a BCC-algebra is a BCKalgebra iff it satisfies the identity (6) xy z = xz y. Methods of construction BCC-algebras from BCK-algebras are given in [3] Mathematics Subject Classification: Primary: 06 F 35, 03 G 25. Key words and phrases: BCC-algebra, BCK-algebra, atom. 81

2 If (G,, 0) is a BCC-algebra, then the relation defined on G by (7) x y iff xy = 0 is a partial order on G with 0 as a smallest element (cf. [2]). Moreover, this relation has the following properties (8) xy zy xz, (9) x y implies xz yz and zy zx. A non-empty subset A of a BCC-algebra G is called a BCK-ideal of G iff 0 A and y, xy A imply x A. A subset B of G is called a BCC-ideal (cf. [4]) iff 0 B and y, xy z B imply xz B. Obviously any BCC-ideal is a BCK-ideal and induces a congruence on a BCC-algebra (cf. [4]), but there are congruences which are not induced by such ideals. In BCK-algebras BCK-ideals are BCC-ideals. 2. Atoms. A non-zero element a G is called an atom of a BCC-algebra G if x a implies x = 0 or x = a. Lemma 1. A non-zero element a G is an atom of G if {0, a} is a BCK-ideal. The converse is not true. Indeed, in a proper BCC-algebra defined by the following table 0 a b c a a b b a 0 0 c c b b 0 an element a is an atom, but {0, a} is not a BCK-ideal. Lemma 2. If every non-zero element of a BCC-algebra G is an atom, then any subalgebra of G is a BCK-ideal. Proof. Let S be a subalgebra and let x, yx S. Since yx y for all x, y G (cf. [2]) and y is an atom of G, then yx = 0 or y = yx S. If yx = 0, then y x which gives y = 0 or y = x. Thus y S, which completes the proof. From the above lammas we obtain Theorem 3. A BCC-algebra contains only atoms iff every its subalgebra is a BCK-ideal. 82

3 Lemma 4. If a b are atoms, then ab = a. Now we consider the set Z(G) = {z G : zx y = zy x for all x, y G}. Obviously, 0 Z(G). Lemma 5. If a Z(G), then a ax x for all x G. Indeed, (a ax)x = ax ax = 0. Theorem 6. Z(G) is a BCK-subalgebra of G. Proof. Let a, b Z(G), x, y G. Then ab (ab y) = ab (ay b) a ay y by (8) and Lemma 5. This, by (9) and (8) gives (ab x)y (ab x)(ab (ab y) = (ax b)(a(ab y) b) ax a(ab y) = a(a(ab y)) x (ab y)x, since a(a(ab y)) ab y by Lemma 5. Thus (ab x)y (ab y)x, which by symmetry gives (ab x)y = (ab y)x. Hence ab Z(G), i.e. Z(G) is a BCC-subalgebra. It is also a BCK-subalgebra because by the definition of Z(G) xy z = xz y for all x, y, z G. Let A(G) denotes the set of all atoms (together with 0) of a BCCalgebra G. Theorem 7. A(G) is a BCK-subalgebra contained in Z(G). Proof. A(G) is non-empty, because 0 G. By Lemma 4 it is also closed with respect to the BCC-operation. We prove A(G) Z(G). Let a A(G). Then for x, y G we have the following four cases: 1 0 a x and a y, 2 0 a x and not(a y), 3 0 not(a x) and a y, 4 0 not(a x) and not(a y). If a x then obviously ax y = 0 for every y G. If not(a x), then ax = a since ax a (by Proposition 2 in [2]) and ax 0. Thus in the case 1 0 we obtain ax y = 0 = ay x, i.e. a Z(G). In the case 2 0 we have ax y = 0 and ay x = ax = 0, which gives ax y = ay x. The case 3 0 is analogous. In the case 4 0 ax y = a = ay x. Hence in any case we obtain ax y = ay x, which proves that A(G) is contained in Z(G). Corollary 8. If all non-zero elements of a given BCC-algebra are atoms then it is a BCK-algebra. As a simple consequence of Lemma 4 we obtain Corollary 9. For any cardinal n 2 there exists only one BCCalgebra in which all non-zero elements are atoms. 83

4 By Theorem 3 we obtain also Corollary 10. Every subalgebra of a BCK-algebra is an BCK-ideal iff every non-zero element of this BCK-algebra is an atom. Finally we note that if a BCK-algebra G has at least one non-atom element, then A(G) Z(G) = G. On the other hand, in some BCCalgebras A(G) = Z(G) G. As an example we consider the algebra G = {0, a, b, c, d, e} defined by the table 0 a b c d e a a a b b b 0 0 a a c c b a 0 a a d d d d d 0 a e e e e e e 0 Since S = {0, a, b, c, d} is a proper BCC-algebra (cf. Table 2 in [1]), then G is a proper BCC-algebra by Proposition 4 in [2] (cf. also Construction 3 in [3]). In this BCC-algebra A(G) = Z(G) = {0, a, e}, but in a subalgebra S we have A(S) = {0, a} = Z(S) = {0, a, d}. Moreover, A(G) is not a BCK-ideal, but A(S) is a BCK-ideal of S and G. Obviously S is a BCC-ideal of G. Note also that Z(S) is not a maximal BCK-subalgebra of S since it not contains a BCK-subalgebra B = {0, a, b}. References [1 ] W.A.Dudek: The number of subalgebras of finite BCC-algebras, Bull. Inst. Math. Academia Sinica, 20(1992), [2 ] W.A.Dudek: On proper BCC-algebras, Bull. Inst. Math. Academia Sinica 20(1992), [3 ] W.A.Dudek: On constructions of BCC-algebras, Selected Papers on BCK- and BCI-algebras, 1(1992), [4 ] W.A.Dudek and X.H.Zhang: On ideals and congruences in BCCalgebras, (to appear). [5 ] Y.Komori: The variety generated by BCC-algebras is finitelly based, Reports Fac. Sci. Shizuoka Univ. 17(1983), [6 ] Y.Komori: The class of BCC-algebras is not a variety, Math. Japonica 29(1984),

5 Wies law A. Dudek Xiaohong Zhang Institute of Mathematics Department of Mathematics Technical University Hanzhong Teachers College Wybrzeże Wyspiańskiego 27 Hanzhong, Shaanxi Province Wroc law Peoples Republic of China Poland 85