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 Iseki [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; less-than-or-equal-to,; *, 1] of type (2, 2, 0), define X by stipulating x X y = y * x and curly less than by putting x curly less than y if and only if y less-than-or-equal-to x, then [S; less-than-or-equal-to,; *, 1] is an implicative commutative semigroup if and only if [S; curly less than,.,X, 1] is a BCK algebra with condition (S); a nonempty subset F of S is an ordered filter of [S; less-than-or-equal-to,.,*, 1] if and only if F is an ideal of [S; curly less than,.,X, 1].