Priestley-style duality for filter-distributive congruential logics
María Esteban, Ramon Jansana
TL;DR
The paper develops a Priestley-style duality for the algebraic counterparts of congruential, finitary, and filter-distributive logics by leveraging optimal logical filters and the \\mathcal{S}-semilattice. It constructs \\mathcal{S}-Priestley spaces as duals to \\mathsf{Alg}\\mathcal{S} and introduces contravariant functors \\mathfrak{O}\\mathrm{p}_{\\mathcal{S}} and \\bullet, with natural isomorphisms establishing a categorical equivalence. The authors characterize how logical properties such as conjunction, disjunction, the deduction-detachment theorem, and inconsistent elements translate into dual-space structures and morphisms, yielding precise dual correspondences (e.g., PC and PDI implications on \\mathcal{S}-Priestley spaces). This framework unifies and extends Priestley-type dualities beyond Boolean and Heyting settings to a wide range of congruential logics, providing a robust tool for semantic and categorical analysis in abstract algebraic logic.
Abstract
We first present a Priestley-style dualitiy for the classes of algebras that are the algebraic counterpart of some congruential, finitary and filter-distributive logic with theorems. Then we analyze which properties of the dual spaces correspond to properties that the logic might enjoy, like the deduction theorem or the existence of a disjunction.