Dedekind-MacNeille and related completions: subfitness, regularity, and Booleanness

TL;DR

The paper investigates when the Dedekind-MacNeille, Bruns-Lakser, ideal, and canonical completions of a bounded distributive lattice $A$ satisfy subfitness, regularity, or Booleanness, using Priestley duality and the proHeyting extension $\mathpzc{p}\mathcal{H} A$ as central tools. It proves key equivalences, such as $\mathcal{DM} A$ being subfit iff $A$ is subfit, and $\mathcal{BL} A$ being Boolean iff $A$ is $\wedge$-subfit, while also establishing that $A^\sigma$ is Boolean exactly when $A$ is Boolean. The work further characterizes regularity and Booleanness via density properties in dual spaces (e.g., $\min X$, $\max X$, Skula topology) and shows how $\mathcal{BL} A$ realizes $\mathcal{DM}(\mathpzc{p}\mathcal{H} A)$, providing a unifying framework across completions. These results illuminate when completions preserve or reflect separation axioms and connect lattice-theoretic properties to dual-space densities, with implications for frame theory and modal semantics.

Abstract

Completions play an important rôle for studying structure by supplying elements that in some sense ``ought to be." Among these, the Dedekind-MacNeille completion is of particular importance. In 1968 Janowitz provided necessary and sufficient conditions for it to be subfit or Boolean. Another natural separation axiom connected to these is regularity. We explore similar characterizations of when closely related completions are subfit, regular, or Boolean. We are mainly interested in the Bruns-Lakser, ideal, and canonical completions, which (unlike the Dedekind-MacNeille completion) satisfy stronger forms of distributivity. The first two are widely used in pointfree topology, while the latter is of crucial importance in the semantics of modal logic.

Figures (4)

• Figure 1: A Priestley space $X$ with ${\sf BL} (X)$ subfit but ${\sf L} (X)$ not.
• Figure 2: A Priestley space $X$ with ${\sf L}(X)$ subfit but ${\sf OpUp}(X)$ not.
• Figure 3: The Priestley space $X$ of $A$.
• Figure 4: A Priestley space $X$ with ${\sf L}(X)$ subfit, ${\sf BL}(X)$ regular, but ${\sf L}(X)$ not regular.

Theorems & Definitions (107)

• Definition 2.1
• Remark 2.2
• Theorem 2.3: Priestley duality
• Remark 2.4
• Remark 2.5
• Theorem 2.7
• Remark 2.8
• Definition 2.9
• Theorem 2.10
• Remark 2.11