Series B: Mathematical Science 31 (1998), pp. 21–25 ANNIHILATORS IN BCK-ALGEBRAS II

Abstract. In this paper we show that for every BCI-algebra X, (1) the family CI(X) of all closed ideal in X forms a pseudo-complemented distributive lattice if and only if X is the BCK-algebra; (2) 8A 2 CI(X)A µ A∗ ∗ () X is the BCK-algebra; 1. BCK-algebra Firstly we define a BCI-algebra and a BCK-algebra. An algebra (X; ¤, 0) of type (2,0) is called a BCI-algebra when it satisfies the conditions: For every x, y, z 2 X, (1) (x ¤ y) ¤ (x ¤ z) · z ¤ y (2) x ¤ (x ¤ y) · y (3) x ¤ x = 0 (4) x ¤ y = y ¤ x = 0 =) x = y,,where the relation ”· ” is defined by x · y if and only if x ¤ y = 0. It is well known that the relation is a partially ordered relation on any BCI-algebra (cf [2]). In BCI-algebras, the following properties hold: (a) x ¤ 0 = x (b) 0 ¤ (x ¤ y) = (0 ¤ x) ¤ (0 ¤ y) (c) (x ¤ y) ¤ z = (x ¤ z) ¤ y (d) (x ¤ z) ¤ (y ¤ z) · x ¤ y (e) x · y =) x ¤ z · y ¤ z (f) x · y =) z ¤ x · z ¤ y We say that a BCI-algebra X is a BCK-algebra if (5) 0 ¤ x = 0 for x 2 X. By an ideal we mean a subset I of X such that it satisfies the conditions