Complete and atomic Tarski algebras

Abstract

Tarski algebras, also known as implication algebras or semi-boolean algebras, are the {→}-subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structuresX, K , where X is a non-empty set and K is non-empty family of subsets of X such that K = X. This duality is a generalization of the known duality between sets and complete and atomic Boolean algebras. We shall also analize the case of complete and atomic Tarski algebras endowed with a complete modal operator, and we will prove a duality for these algebras.