Canonical extensions of locally compact frames

TL;DR

This paper extends the concept of canonical completions from ordered structures to frames, showing how locally compact frames embed into completely distributive lattices and enabling frame-theoretic representation of monotone maps.

Contribution

It adapts algebraic techniques to frame theory, generalizing canonical extensions and providing new insights into their structure and applications.

Findings

Locally compact frames embed into completely distributive lattices.

The construction generalizes canonical extensions for distributive and proximity lattices.

Enables frame-theoretic representation of monotone maps.

Abstract

Canonical extension of finitary ordered structures such as lattices, posets, proximity lattices, etc., is a certain completion which entirely describes the topological dual of the ordered structure and it does so in a purely algebraic and choice-free way. We adapt the general algebraic technique that constructs them to the theory of frames. As a result, we show that every locally compact frame embeds into a completely distributive lattice by a construction which generalises, among others, the canonical extensions for distributive lattices and proximity lattices. This construction also provides a new description of a construction by Marcel Ern\'e. Moreover, canonical extensions of frames enable us to frame-theoretically represent monotone maps with respect to the specialisation order.