A fresh perspective on canonical extensions for bounded lattices

TL;DR

This paper introduces a new, duality-based method for constructing canonical extensions of bounded lattices, unifying the approach for all such lattices and ensuring functoriality.

Contribution

It develops a functorial construction of canonical extensions for all bounded lattices using duality theory, extending previous methods from distributive lattices.

Findings

Constructs a canonical extension functor valid for all bounded lattices.

Shows the functor decomposes into two hom-functor-like components.

Provides a duality-based framework inspired by Ploscica and Allwein-Hartonas representations.

Abstract

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of thetheory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Ploscica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphismsthe Ploscica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Ploscica's paper. This leads to a construction of canonical extension valid for all bounded lattices,which is shown to be functorial, with the property that the canonical extension functor decomposes asthe composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors.