Topological duality for Tarski algebras. (English) Zbl 1136.03046

Celani, Sergio A.; Cabrer, Leonardo M.

Algebra Univers. 58, No. 1, 73-94 (2008).

Summary: In this paper we generalize the representation theory developed for finite Tarski algebras given in [S. A. Celani, “Modal Tarski algebras”, Rep. Math. Logic 39, 113–126 (2005; Zbl 1105.03069)]. We introduce the notion of Tarski space as a generalization of the notion of dense Tarski set, and we prove that the category of Tarski algebras with semi-homomorphisms is dually equivalent to the category of Tarski spaces with certain closed relations, called \(T\)-relations. By these results we obtain that the algebraic category of Tarski algebras is dually equivalent to the category of Tarski spaces with certain partial functions. We apply these results to give a topological characterization of the subalgebras.

MSC:

03G25	Other algebras related to logic
06E15	Stone spaces (Boolean spaces) and related structures

Keywords:

Tarski algebras; representation theorem; topological duality; subalgebras

Citations:

Zbl 1105.03069