C(S) extensions of S-I-BCK-algebras

In this paper we consider more systematically the centralizer C(S) of the set <TEX>$S = {f_a <TEX>$mid$</TEX> f_a : X o X ; x longmapsto x * a, a in X}$</TEX> with respect to the semigroup End(X) of all endomorphisms of an implicative BCK-algebra X with the condition (S). We obtain a series of interesting results. The main results are stated as follows : (1) C(S) with repect to a binary operation * defined in a certain way forms a bounded implicative BCK-algebra with the condition (S). (2) X can be imbedded in C(S) such that X is an ideal of C(S)/ (3) If X is not bounded, it can be imbedded in a bounded subalgebra T of C(S) such that X is a maximal ideal of T. (4) If $X ( eq {0})$ is semisimple, C(S) is BCK-isomorphic to <TEX>$prod_{i in I}{A_i}$</TEX> in which <TEX>${A_i}_{i in I}$</TEX> is simple ideal family of X.