Compatible Deductive Systems of Pulexes

The notion of (compatible) deductive system of a pulex is defined and some properties of deductive systems are investigated. We also define a congruence relation on a pulex and show that there is a bijective correspondence between the compatible deductive systems and the congruence relations. We define the quotient algebra induced by a compatible deductive system and study its properties. 1. Introduction 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 pseudo-BCK 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 , where is a poset with the greatest element and , are binary operations on such that, for all , , , we have(1) , ,(2) , ,(3) iff iff . Theorem 2 (see [11]). An algebra of type is a pseudo-BCK algebra if and only if it satisfies the following:(1) and ;(2) , ;(3) , ;(4) and implies , the same for . Definition 3 (see [11]). If the partial order of a pseudo-BCK algebra is a lattice order, with the lattice operations and , then is said to be a pseudo-BCK lattice and it will be denoted by . Example 4. Let such that , , and are incomparable with . The operations and are given as follows: Then is a pseudo-BCK lattice. Definition 5 (see [9]). A pulex is a structure such that(1) is a