In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1

Achim Jung and Drew Moshier developed a Stone-type duality theory for bitopological spaces, amongst others, as a practical tool for solving a particular problem in the theory of stably compact spaces. By doing so they discovered that the duality of bitopological spaces and their algebraic counterparts, called d-frames, covers several of the known dualities. In this thesis we aim to take Jung's and Moshier's work as a starting point and fill in some of the missing aspects of the theory. In particular, we investigate basic categorical properties of d-frames, we give a Vietoris construction for d-frames which generalises the corresponding known Vietoris constructions for other categories, and we investigate the connection between bispaces and a paraconsistent logic and then develop a suitable (geometric) logic for d-frames.

This thesis is divided into four chapters. In the first one, all the pre-requisite theory of semigroups and groupoids is introduced, as well as a few new results - such as a short study of ∨-ideals and quotients in distributive semigroups and a non-commutative Loomis-Sikorski Theorem. In the second chapter, we motivate and describe the sofic property for probability measure-preserving groupoids and prove several permanence properties for the class of sofic groupoids. This provides a common ground for similar results in the particular cases of groups and equivalence relations. In particular, we prove that soficity is preserved under finite index extensions of groupoids. We also prove that soficity can be determined in terms of the full group alone, answering a question by Conley, Kechris and Tucker-Drob. In the third chapter we turn to the classical problem of reconstructing a topological space from a suitable structure on the space of continuous functions. We prove that a locally compact Hausdorff space can be recovered from classes of functions with values on a Hausdorff space together with an appropriate notion of disjointness, as long as some natural regularity hypotheses are satisfied. This allows us to recover (and even generalize) classical theorem by Kaplansky, Milgram, Banach-Stone, among others, as well as recent results of the similar nature, and obtain new consequences as well. Furthermore, we extend the techniques used here to obtain structural theorems related to topological groupoids. In the fourth and final chapter, we study dynamical aspects of partial actions of inverse semigroups, and in particular how to construct groupoids of germs and (partial) crossed products and how do they relate to each other. This chapter is based on joint work with Viviane Beuter.

Le thème central de la présente thèse est le contenu sémantique des quantificateurs logiques. Dans leur forme la plus simple, les quantificateurs permettent d’établir l’existence, ou la non-existence, d’individus répondant à une propriété. En tant que tels, ils incarnent la richesse et la complexité de la logique du premier ordre, par delà la logique propositionnelle. Nous contribuons à l’analyse sémantique des quantificateurs, du point de vue de la théorie de la dualité, dans trois domaines différents des mathématiques et de l’informatique théorique. D’une part, dans la théorie des langages formels à travers la logique sur les mots. D’autre part, dans la logique intuitionniste propositionnelle et dans l’étude de l’interpolation uniforme. Enfin, dans la topologie catégorique et dans la sémantique catégorique de la logique du premier ordre. / The unifying theme of the thesis is the semantic meaning of logical quantifiers. In their basic form quantifiers allow to state theexistence, or non-existence, of individuals satisfying a property. As such, they encode the richness and the complexity of predicate logic, as opposed to propositional logic. We contribute to the semantic understanding of quantifiers, from the viewpoint of duality theory, in three different areas of mathematics and theoretical computer science. First, in formal language theory through the syntactic approach provided by logic on words. Second, in intuitionistic propositional logic and in the study of uniform interpolation. Third, in categorical topology and categorical semantics for predicate logic.