Logic, Algebra and Topology. Investigations into canonical extensions, duality theory and point-free topology. Jacob Vosmaer Abstract: In this dissertation we discuss three subjects: canonical extensions of lattice-based algebras, Stone duality for distrbutive lattices with operators, and a generalization of the point-free Vietoris powerlocale construction. In Chapters 2 and 3, we study canonical extensions of lattice-based algebras in relation to topological algebra, profinite completions and directed complete partial orders (dcpo’s). We provide a topological characterization theorem for the canonical extension of a lattice in §2.1.3, and we give an improved characterization of canonical extensions of order-preserving maps as maximal continuous extensions, along with further continuitresults, in §2.2. The improvement in the results of §2.2 lies in the fact that they hold for arbitrary rather than distributive lattices. In §2.3, we show how canonical extensions of lattices can be characterized using dcpo presentations, concluding Chapter 2. In Chapter 3 we discuss canonical extensions of arbitrary maps and canonical extensions of lattice-based algebras, both in relation to topological algebra. In §3.3.2, we show that the canonical extenion of a surjective lattice-based algebra homomorphism is again an algebra homomorphism. We use this fact to show in §3.4.1 that the profinite completion of any lattice-based algebra A can be characterized as a complete quotient of the canonical extension of A. Subsequently, in §3.4.2, we investigate necessary and sufficient circumstances for the profinite completion of A to be equal to the canonical extension of A. We conclude Chapter 3 with a discussion of a universal property of canonical extensions with respect to Boolean topological algebras in §3.4.3. In Chapter 4, we study discrete Stone duality for semi-topological distributive lattices with operators (DLO’s) and ordered Kripke frames. In §4.1, we study the duality between profinite DLO’s and the corresponding hereditarily finite ordered Kripke frames. We consider special cases of this duality in §4.2, namely distributive lattices, Boolean algebras, Heyting algebras and modal algebras. Finally, in §4.3, we show that if we restrict our attention to Boolean algebras with operators (BAO’s) rather than DLO’s, we can characterize not only profinite BAO’s via Stone duality, but also compact Hausdorff and Boolean topological BAO’s. We show that compact Hausdorff BAO’s (and Boolean topological BAO’s) are dual to image-finite Kripke frames. We use this knowledge to study the embedding of Kripke frames into their ultrafilter extensions in §4.3.2. In Chapter 5, we use a geometric version of the Carioca axioms for coalgebraic modal logic with the cover modality to give a new description of the point-free Vietoris construction. In §5.3.1 we introduce the T-powerlocale construction, where T : Set → Set is a weak-pullback preserving, standard, finitary endofunctor on the category of sets. We then go on to show that the P-powerlocale, where P is the covariant powerset functor, is the usual Vietoris powerlocale in §5.3.3. In §5.3.4 we show that the T-powerlocale construction yields a functor VT on the category of frames, and we show how to lift natural transformations between set functors T′ and T to natural transformations between T-powerlocale functors VT and VT ′ . In §5.3.5 we show that the T -powerlocale can be presented using a flat site presentation rather than an frame presentation; this gives us an algebraic proof for the fact that formulas in our geometric coalgebraic modal logic have a disjunctive normal form. Finally in §5.4, we show that the T-powerlocale construction preserves regularity, zero-dimensionality and the combination of zero-dimensionality and compactness.