Publications

In 'Topological Kripke models' (1974), 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 combining coalgebraic modal logic and the Vietoris functor

This paper introduces an endofunctor VT on the category of frames, parametrized by an endofunctor T on the category Set that satisfies certain constraints. This generalizes Johnstone’s construction of the Vietoris powerlocale, in the sense that his construction is obtained by taking for T the finite covariant power set functor. Our construction of the T-powerlocale VT L out of a frame L is based on ideas from coalgebraic logic and makes explicit the connection between the Vietoris construction and Moss’s coalgebraic cover modality.

We show how to extend certain natural transformations between set functors to natural transformations between T-powerlocale functors. Finally, we prove that the operation VT preserves some properties of frames, such as regularity, zero-dimensionality, and the combination of zerodimensionality and compactness.

We show how canonical extensions of distributive lattices with operators (DLOs) can be presented as DCPO algebras, using machinery developed by Jung e.a. (2008). Moreover, we show that a well-known canonicity result for DLOs be seen as a special case of a result by Jung e.a. (2008).

These topics are also discussed in my PhD dissertation.

An overview of the state of the art in canonical extensions. Provides a different perspective on some of the results in my PhD dissertation.

Ch. 1: Introduction.

Ch. 2: Canonical extensions: a domain theoretic approach. Characterization of the canonical extension of a lattice using both topology and dcpo-presentations. Topological and universal properties of canonical extensions of order-preserving maps between lattices.

Ch. 3: Canonical extensions and topological algebra. Canonical extensions of lattice-based algebras in relation to the profinite completion. Extensions of arbitrary maps between lattices, under the condition "HSP(L) is a finitely generated variety". Universal properties of the canonical extension with respect to topological algebras.

Ch. 4: Duality, profiniteness and completions. Duality for profinite distributive lattices with operators. Duality between compact Hausdorff zero-dimensional topological Boolean algebras with operators and image-finite Kripke frames; relation between ultrafilter extensions of Kripke frames and reflexivity of the canonical extension.

Ch. 5: Coalgebraic modal logic in point-free topology. Using coalgebraic modal logic with the "nabla" modality to describe and generalize the point-free Vietoris construction.

We show that a particular modal logic introduced by Thomason in 1974 is incomplete with respect to the class of all complete modal algebras.

We investigate under what circumstances the completions listed in the title differ or coincide, for the cases of distributive lattices, Heyting algebras and Boolean algebras, using Stone duality.

A precursor of Chapters 3 and 4 of my PhD dissertation, focussing mainly on modal algebras.

A student paper in which we characterize the essentially Σ-1 formulas of ΣL, which is an extension of provability logic in which one can also express whether a formula is Σ-1.