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.