On algebras with a generalized implication

摘要

We introduce the notion of gi-algebra as a generalization of dual BCK-algebra, and define the notions of strong, commutative and transitive gi-algebra, and then we show that an interval a?‘l = {a a?? P | l a‰¤ a} in a strong and commutative gi-algebra P is a lattice. Also, we define a congruence relation a??D on a transitive gi-algebra P and show that the quotient set P/a??D is a gi-algebra and a dual BCK-algebra.