Duality Theory for Sequent Algebras

Abstract

Sequent algebras were introduced by D. Vakarelov \cite{VakarelovSequentALg} as a generalization of contact algebras, replacing the binary contact relation with a relation between finite sets of regions and regions. In this paper, we present sequent algebras as Boolean algebras endowed with a finitary closure operator satisfying four conditions. We study the irreducible closed sets of these operators, which coincide with Vakarelov's clans, and use them to develop a topological representation theory for sequent algebras. We introduce the categories of sequent algebras and sequent spaces and establish a dual equivalence between them, as well as a restricted dual equivalence for morphisms preserving the full Boolean structure. We also characterize sequent congruences in terms of positive filters and minimally decreasing closed subsets of the associated sequent spaces. Finally, we show that the category of sequent algebras with sequent homomorphisms is topological over the category of Boolean algebras, obtaining completeness and cocompleteness and a complete lattice structure on the sequent structures over each Boolean algebra.