Distributive envelopes and topological duality for lattices via canonical extensions

TL;DR

This paper develops a topological duality for bounded lattices that extends Stone duality, introduces a universal construction for distributive lattice envelopes, and links these to completions of quasi-uniform spaces, providing a new perspective on lattice duality.

Contribution

It introduces a novel topological duality for bounded lattices with non-standard morphisms and a universal construction for distributive envelopes, extending classical dualities.

Findings

Duality generalizes Stone duality for bounded distributive lattices.

Distributive envelopes of a lattice relate to completions of quasi-uniform spaces.

The dual spaces of envelopes coincide with certain space completions.

Abstract

We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of 'admissibility' to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasi-uniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.