
Algebra 2013
Commutative and Bounded BEalgebrasDOI: 10.1155/2013/473714 Abstract: We introduce the notions of the commutative and bounded BEAlgebras. We give some related properties of them. 1. Introduction Imai and Iséki introduced two classes of abstract algebras called BCKalgebras and BCIalgebras [1, 2]. It is known that the class of BCKalgebras is a proper subclass of BCIalgebras. In [3, 4], Hu and Li introduced a wide class of abstract algebras called BCHalgebras. They have shown that the class of BCIalgebras is a proper subclass of BCHalgebras. Neggers and Kim [5] introduced the notion of dalgebras which is another generalization of BCKalgebras, and also they introduced the notion of Balgebras [6, 7]. Jun et al. [8] introduced a new notion called BHalgebra which is another generalization of BCH/BCI/BCKalgebras. Walendziak obtained some equivalent axioms for Balgebras [9]. C. B. Kim and H. S. Kim [10] introduced the notion of BMalgebra which is a specialization of Balgebras. They proved that the class of BMalgebras is a proper subclass of Balgebras and also showed that a BMalgebra is equivalent to a 0commutative Balgebra. In [11], H. S. Kim and Y. H. Kim introduced the notion of BEalgebra as a generalization of a BCKalgebra. Using the notion of upper sets they gave an equivalent condition of the filter in BEalgebras. In [12, 13], Ahn and So introduced the notion of ideals in BEalgebras and proved several characterizations of such ideals. Also they generalized the notion of upper sets in BEalgebras and discussed some properties of the characterizations of generalized upper sets related to the structure of ideals in transitive and selfdistributive BEalgebras. In [14], Ahn et al. introduced the notion of terminal section of BEalgebras and provided the characterization of the commutative BEalgebras. In this paper we introduce the notion of bounded BEalgebras and investigate some properties of them. 2. Preliminaries Definition 1 (see [11]). An algebra of type (2, 0) is called a BEalgebra 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 BEalgebra. Definition 3. A BEalgebra is said to be selfdistributive if for all , , and . Example 4 (see [11]). Let be a set with the following table: Then is a selfdistributive BEalgebra. Proposition 5 (see [14]). Let be a selfdistributive BEalgebra. If , then, for all , , and in , the following inequalities hold:(i) ,(ii) . Definition 6 (see [15]). A dual BCKalgebra is an algebra of type (2,0) satisfying (BE1) and (BE2) and the
