On algebras with a generalized implication

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 ↑l = {a ∈ P | l ≤ a} in a strong and commutative gi-algebra P is a lattice. Also, we define a congruence relation ～ _D on a transitive gi-algebra P and show that the quotient set P/～ _D is a gi-algebra and a dual BCK-algebra.