Degrees of join-distributivity via Bruns-Lakser towers

TL;DR

This work introduces a cardinally indexed framework for measuring join-distributivity in bounded distributive lattices by deploying Bruns-Lakser towers inside completions. By intertwining the Bruns-Lakser completion $\mathcal{BL}A$, the Dedekind-MacNeille completion $\mathcal{DM}A$, the proHeyting extension $\mathpzc{p}\mathcal{H}A$, and Priestley duality, the authors define $\mathcal{BL}_\kappa$-towers and prove characterizations of $\kappa$-frames, $\kappa$-distributivity, and related hierarchies (e.g., $\kappa$-proHeyting, $\kappa$-CH) via collapse equalities like $\mathcal{BL}_\kappa A = A$ or $\mathcal{BL}_\kappa(\mathcal{DM}A) = \mathcal{DM}A$. The results yield dual descriptions of these hierarchies and a natural generalization of Esakia representations to proHeyting lattices, alongside concrete ladder-based counterexamples that separate the various $\kappa$-classes. These constructions provide a fine-grained, cardinality-sensitive view of how distributivity interacts with completeness and duality in lattice theory. The Funayama-type examples illustrate that BL and DM completions can commute in the absence of proHeyting, while κ-generalizations reveal intrinsic non-containment phenomena among the hierarchies.

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 $κ$-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.

Figures (7)

• Figure 1: $(\omega\text{-ladder})\oplus\omega^{\mathrm{op}}$
• Figure 2: BL-towers of $A \leq B$
• Figure 3: BL-towers of $A$, $\mathcal{DM} A$, and $\mathpzc{p}\mathcal{H} A$
• Figure 4: $\lambda$-ladder with top
• Figure 5: Linear sum $\lambda\oplus\omega^{\mathrm{op}}$

Theorems & Definitions (98)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Proposition 2.6
• Remark 2.7
• Definition 2.8
• Proposition 2.9
• proof
• Definition 2.10