Compatible Deductive Systems of Pulexes

Imai and Iséki [1] introduced the concept of BCK-algebra as a generalization of notions of set difference operation and propositional calculus. The notion of pseudo-BCK algebra was introduced by Georgescu and Iorgulescu [2] as generalization of BCK-algebras not assuming commutativity. Hájek [3] introduced the concept of BL-algebras as the general semantics of basic fuzzy logic (BL-logic). Iorgulescu studied BCK-algebras and their relation to BL-algebras [4, 5]. BL-algebra has been generalized in different ways [6– 8]. Hájek in [9] introduced flea-algebra as a generalization of BL-algebra. He proved that the implication reduct of a flea-algebra is a pseudo-BCK algebra with three additional conditions. The pseudo-BCK algebra with these conditions is called pulex. In this section some preliminary definitions and theorems are stated. In Section 2, we introduce the notions of deductive system, lattice filter, and subalgebra of pulexes and obtain some properties which are not true in a pseudoBCK lattice in general. In Section 3, we define concepts of compatible deductive system and congruence relation on a pulex and show that there is a correspondence between the set of all compatible deductive systems of a pulex and the set of all congruence relations on a pulex which is not true in a pseudo-BCK lattice. After that we prove that the quotient algebra defined by a congruence relation is a pulex and we obtain some related results. Definition 1 (see [10]). A pseudo-BCK algebra is a structure A = (A, ≤, → , 󴁄󴀼, 1), where (A, ≤) is a poset with the greatest element 1 and → , 󴁄󴀼 are binary operations on A such that, for all x, y, z ∈ A, we have