Universal stone duality via the concept of topological dualizability and its applications to many-valued logic

Abstract

We propose the concept of topological dualizability as the condition of possibility of Stone duality, and thereby give a non-Hausdorff extension of the primal duality theorem in natural duality theory in universal algebra. The primal duality theorem is a vast generalization of the classic Stone duality for Boolean algebras, telling that any varieties generated by functionally complete algebras, such as the algebras of Emil Post's finite-valued logics, are categorically equivalent to zero-dimensional compact Hausdorff spaces. Here we show a non-Hausdorff extension of primal duality: any varieties generated by certain weakly functionally complete or topologically dualizable algebras are categorically dually equivalent to coherent spaces, a special class of compact sober spaces. This generalizes the Stone duality for distributive lattices and Heyting algebras (as a subclass of distributive lattices) in the spirit of primal duality theory. And we give applications of the general theorem to algebras of Łukasiewicz many-valued logics. The concept of topological dualizability is arguably the key to the universal algebraic unification of Stone-type dualities; in the present paper, we take the first steps in demonstrating this thesis.