We prove a categorical duality between class of abstract algebras partial functions and (small) topological categories. The are the isomorphs collections closed under operations composition, antidomain, range, preferential union (or ‘override’). categories those whose space objects is Stone space, source map local homeomorphism, target open, all arrows epimorphisms.

The Schützenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Schützenberger product to the level of monoids in an algebraic category D , leading to a uniform view of the corresponding constructions for monoids (Schützenberger), ordered monoids (Pin), idempotent semirings (Klíma and Polák) and algebras over a field (Re...

This paper is a contribution to the theory of what might be termed $0$-dimensional non-commutative spaces. We prove that associated with each inverse semigroup $S$ Boolean presented by abstract versions Cuntz-Krieger relations. call this Exel completion and show it arises from Exel's tight groupoid under Stone duality.

In [17], Esakia uses the Vietoris topology to give a coalgebra-flavored definition of topological Kripke frames, thus relating the Vietoris topology, modal logic and coalgebra. In this chapter, we sketch some of the thematically related mathematical developments that followed. Specifically, we look at Stone duality for the Vietoris hyperspace and the Vietoris powerlocale, and at recent work com...

By Gelfand-Neumark duality, the category C∗Alg of commutative C∗algebras is dually equivalent to the category of compact Hausdorff spaces, which by Stone duality, is also dually equivalent to the category uba` of uniformly complete bounded Archimedean `-algebras. Consequently, C∗Alg is equivalent to uba`, and this equivalence can be described through complexification. In this article we study u...

Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general categorical dualities the basis and closure conditions, which encompasses variety for topological spaces, convex measurable spaces (some are new duality results their own). It works as method generate analogues between frames (locales) beyond topology, e.g., spaces. After establishing t...