Logics for Compact Hausdorff Spaces via de Vries Duality
Thomas Santoli
Abstract:
In this thesis, we introduce a finitary logic which is sound and complete with respect to de Vries algebras, and hence by de Vries duality, this logic can be regarded as logic of compact Hausdorff spaces. In order to achieve this, we first introduce a system S which is sound and complete with respect to a wider class of algebras. We will also define Π2 -rules and establish a connection between Π2-rules and inductive classes of algebras, and we provide a criterion for establishing when a given Π2-rule is admissible in S. Finally, by adding two particular rules to the system S, we obtain a logic which is sound and complete with respect to de Vries algebras. We also show that these two rules are admissible in S, hence S itself can be regarded as the logic of compact Hausdorff spaces. Moreover, we define Sahlqvist formulas and rules for our language, and we give Sahlqvist correspondence results with respect to semantics in pairs (X, R) where X is a Stone space and R a closed binary relation. We will compare this work with existing literature.