On cyclic B-algebras

Abstract

In this paper, the notion of cyclic B-algebras is introduced and some of its properties are investigated. Also, the laws of exponents for Balgebras are determined. Lastly, this paper seeks to confirm whether the class of cyclic B-algebras and the class of cyclic groups coincide.

Some properties of cyclic B-algebras

“…From [4], if S is a subset of X, then S B is the intersection of all subalgebra H of X such that S ⊆ H, and the subalgebra S B of X is called the subalgebra generated by S. If X = S B , then S is called a set of generators for X.…”

“…Remark 2.1 [4] Let S be a subset of X. Then S B is the smallest subalgebra of X containing S. If either S = ∅ or S = {0}, then S B = {0}.…”

“…The m and n in the above theorem are positive integers. In [4], for each x ∈ X, N.C. Gonzaga and J.P. Vilela defined −x = 0 * x and x −n = (−x) n for each n ≥ 1. Now, for the purpose of this paper, we define…”

“…There are several properties of B-algebras as established by some authors [1−10]. In [4], N.C. Gonzaga and J.P. Vilela introduced cyclic B-algebras. They also established the laws of exponents for B-algebras.…”

Analyzing Some Structural Properties of Topological B-Algebras

“…Thus, certain properties and results on -algebras were established via the properties of groups. Readers may refer to [4][5][6][7][8] for some of these "group-alike" properties and results on -algebras.…”