Provability, Computability and Reflection.

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., structures \, where X is a non-empty set and \ is non-empty family of subsets of X such that \. 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. (shrink)

In this paper we shall study the representation theory for Tarski algebras with a modal operator. In particular, we shall give a representation for finite Tarski algebras and finite Tarski algebras with a modal operator by means of the so-called Tarski sets and Tarski relational sets, respectively. We will also consider some varieties of Tarski modal algebras.