Implicative commutative semigroups are equivalent to a class of BCK algebras

Chan and Shum 2] introduced the notion of implicative semigroups and obtained some of its important properties. BCK algebras with condition (S) were introduced by Iséki 4] and extensively investigated by several authors. In this note, we prove that implicative commutative semigroups are equivalent to BCK algebras with condition (S), that is, given an algebra <S;≤,·,*,1> of type (2,2,0), define ⊗ by stipulatingx⊗y=y*x and ≺ by puttingx≺y if and only ify≤x, then <S≤,·,*,1> is an implicative commutative semigroup if and only if <S;≺,·,⊗, 1> is a BCK algebra with condition (S); a nonempty subsetF ofS is an ordered filter of <S;≤,·,*, 1> if and only ifF is an ideal of <S;≺,·, ⊗, 1>. The author would like to thank the referee for his valuable comments which helped in the modification of this paper.