On topological basic algebras

Abstract

In this paper, the notions of (semi) topological basic algebra and (semi) topological implication basic algebra are introduced, along with evaluating their properties. Then, different operations are defined based on basic algebras and the relationship between semicontinuity and continuity of operations is considered. In addition, the separation axioms on (semi) topological basic algebras are investigated by considering some conditions implying that a (semi) topological basic algebra becomes a T _i - space, for i ∈ {0, 1, 2}. In the sequel, some relations between (weak) ideals and (weak) filters of basic algebras are obtained and (left) topological (implication) basic algebra is constructed by using the concepts of (weak) filters, which is a zero dimensional, normal, disconnected, locally compact and completely regular (left) topological space. Further, the notion of quotient basic algebras are presented along with evaluating the interaction of topological basic algebras and topological quotient basic algebras. Finally, it is proved that there is an implication basic algebra ${\mathcal{IB}}^{*}$ with cardinality n + 1 and filter F^* = F ∪ {z^*}, which $z^{*}\notin \mathcal{IB}$ for any implication basic algebra $\mathcal{IB}$ of cardinality n and filter F. Accordingly, it is proved that there is at least one nontrivial regular and normal topological implication basic algebra of cardinality n.