Remarks on affine complete distributive lattices

TL;DR

This paper characterizes affine complete distributive lattices via Priestley spaces, proves closure properties, and shows that any distributive lattice embeds into an affine complete one, with ap[0,1] being initial in this class.

Contribution

It provides a Priestley space characterization of affine complete lattices and establishes their closure properties and embedding capabilities.

Findings

Affine complete lattices are closed under products and free products.

Every distributive lattice embeds into an affine complete lattice.

ap[0,1] is initial in the class of affine complete lattices.

Abstract

We characterise the Priestley spaces corresponding to affine complete bounded distributive lattices. Moreover we prove that the class of affine complete bounded distributive lattices is closed under products and free products. We show that every (not necessarily bounded) distributive lattice can be embedded in an affine complete one and that $\mathbb{Q} \cap [0,1]$ is initial in the class of affine complete lattices.