Canonical extensions of locally compact frames
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.
Cite
@article{arxiv.1910.03542,
title = {Canonical extensions of locally compact frames},
author = {Tomáš Jakl},
journal= {arXiv preprint arXiv:1910.03542},
year = {2022}
}