Discussiones Mathematicae ser. Algebra and Stochastic Methods 15 (1995), 81 85 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]. 1991 Mathematics Subject Classification: Primary: 06 F 35, 03 G 25. Key words and phrases: BCC-algebra, BCK-algebra, atom. 81
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 0 0 0 0 0 a a 0 0 0 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
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
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 0 0 0 0 0 0 0 a a 0 0 0 0 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), 129-136. [2 ] W.A.Dudek: On proper BCC-algebras, Bull. Inst. Math. Academia Sinica 20(1992), 137-150. [3 ] W.A.Dudek: On constructions of BCC-algebras, Selected Papers on BCK- and BCI-algebras, 1(1992), 93-96. [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), 13-16. [6 ] Y.Komori: The class of BCC-algebras is not a variety, Math. Japonica 29(1984), 391-394. 84
Wies law A. Dudek Xiaohong Zhang Institute of Mathematics Department of Mathematics Technical University Hanzhong Teachers College Wybrzeże Wyspiańskiego 27 Hanzhong, Shaanxi Province 50-370 Wroc law Peoples Republic of China Poland e-mail: dudek@math.im.pwr.wroc.pl. 85