Degrees of join-distributivity via Bruns-Lakser towers

TL;DR

This paper introduces a hierarchy of degrees of join-distributivity in bounded distributive lattices using Bruns-Lakser towers and Priestley duality, generalizing Esakia's representation of Heyting lattices.

Contribution

It develops a new hierarchy of join-distributivity degrees via Bruns-Lakser towers and extends duality-based characterizations to proHeyting lattices.

Findings

Established hierarchies measuring $6$-degrees of distributivity.

Provided dual characterizations using Priestley duality.

Generalized Esakia's representation to proHeyting lattices.

Abstract

We utilize the Bruns-Lakser completion to introduce Bruns-Lakser towers of a meet-semilattice. This machinery enables us to develop various hierarchies inside the class of bounded distributive lattices, which measure $\kappa$-degrees of distributivity of bounded distributive lattices and their Dedekind-MacNeille completions. We also use Priestley duality to obtain a dual characterization of the resulting hierarchies. Among other things, this yields a natural generalization of Esakia's representation of Heyting lattices to proHeyting lattices.