Commutative and Bounded BE-algebras

We introduce the notions of the commutative and bounded BE-Algebras. We give some related properties of them. 1. Introduction Imai and Iséki introduced two classes of abstract algebras called BCK-algebras and BCI-algebras [1, 2]. It is known that the class of BCK-algebras is a proper subclass of BCI-algebras. In [3, 4], Hu and Li introduced a wide class of abstract algebras called BCH-algebras. They have shown that the class of BCI-algebras is a proper subclass of BCH-algebras. Neggers and Kim [5] introduced the notion of d-algebras which is another generalization of BCK-algebras, and also they introduced the notion of B-algebras [6, 7]. Jun et al. [8] introduced a new notion called BH-algebra which is another generalization of BCH/BCI/BCK-algebras. Walendziak obtained some equivalent axioms for B-algebras [9]. C. B. Kim and H. S. Kim [10] introduced the notion of BM-algebra which is a specialization of B-algebras. They proved that the class of BM-algebras is a proper subclass of B-algebras and also showed that a BM-algebra is equivalent to a 0-commutative B-algebra. In [11], H. S. Kim and Y. H. Kim introduced the notion of BE-algebra as a generalization of a BCK-algebra. Using the notion of upper sets they gave an equivalent condition of the filter in BE-algebras. In [12, 13], Ahn and So introduced the notion of ideals in BE-algebras and proved several characterizations of such ideals. Also they generalized the notion of upper sets in BE-algebras and discussed some properties of the characterizations of generalized upper sets related to the structure of ideals in transitive and self-distributive BE-algebras. In [14], Ahn et al. introduced the notion of terminal section of BE-algebras and provided the characterization of the commutative BE-algebras. In this paper we introduce the notion of bounded BE-algebras and investigate some properties of them. 2. Preliminaries Definition 1 (see [11]). An algebra of type (2, 0) is called a BE-algebra if, for all , , and in ,(BE1) ,(BE2) ,(BE3) ,(BE4) . In , a binary relation “ ” is defined by if and only if . Example 2 (see [11]). Let be a set with the following table: Then is a BE-algebra. Definition 3. A BE-algebra is said to be self-distributive if for all , , and . Example 4 (see [11]). Let be a set with the following table: Then is a self-distributive BE-algebra. Proposition 5 (see [14]). Let be a self-distributive BE-algebra. If , then, for all , , and in , the following inequalities hold:(i) ,(ii) . Definition 6 (see [15]). A dual BCK-algebra is an algebra of type (2,0) satisfying (BE1) and (BE2) and the