Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because — in addition to elegant abstraction — they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality...

Let CABA be the category of complete and atomic boolean algebras homomorphisms, let CSL meet-semilattices meet-homomorphisms. We show that forgetful functor from to has a left adjoint. This allows us describe an endofunctor H on such Alg(H) for is dually equivalent Coalg(P) coalgebras powerset P Set. As consequence, we derive Thomason duality Tarski duality, thus paralleling how J\'onsson-Tarsk...

We present a duality theorem for bounded lattices that improves and strengthens Urquhart's topological representation for lattices. Rather than using maximal, disjoint lter-ideal pairs, as Urquhart does, we use all disjoint lter-ideal pairs. This allows not only for establishing a bijective correspondance between lattices and a certain kind of doubly ordered Stone Spaces (Urquhart), but for a f...

Duality theory emerged from the work by Marshall Stone [18] on Boolean algebras and distributive lattices in the 1930s. Later in the early 1970s Larisa Maksimova [10, 11] and Hilary Priestley [15, 16] developed analogous results for Heyting algebras, topological Boolean algebras, and distributive lattices. Duality for bounded, not necessarily distributive lattices, was developed by Alstir Urquh...

We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective étale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with in...

In this paper we relate two generalisations of the finite monoid recognisers of automata theory for the study of circuit complexity classes: Boolean spaces with internal monoids and typed monoids. Using the setting of stamps, this allows us to generalise a number of results from algebraic automata theory as it relates to Büchi’s logic on words. We obtain an Eilenberg theorem, a substitution pri...

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The g...

Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness...

The Stone dualities for accessible categories and the subclass of Diers categories provide limit and colimit structuring principles for query languages and the associated database. As a motivating example, we consider relational databases. Relational databases are given by database schemata, the syntax, and the class of sets of relations satisfying the syntactic constraints, called instances. R...