Annihilators on weakly standard BCC-algebras

BCK-algebras, introduced in the 1960s by Imai and Iséki [3], form a well-known class of algebras intensively studied in algebraic logic during the last decades. The reason of their introduction was practically twofold: they describe general properties of algebras of sets with the set subtraction as a binary operation, and secondly, they form a natural generalization of algebraic counterparts of implicational reducts of several kinds of logics. Namely, the class of implication algebras, introduced by Abbott when describing algebraic properties of the logical connective implication in a classical propositional logic, is a proper subclass of BCK-algebras. Bounded commutative BCK-algebras are known to be equivalent to a class of MV-algebras, arising in a fuzzy logic. In connection with a problem whether the class of BCK-algebras forms a variety, Komori [4] introduced and studied a wider class of BCC-algebras. We start with the following axiomatic system.