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

TL;DR

This paper investigates conditions under which various lattice completions, including Dedekind-MacNeille, Bruns-Lakser, ideal, and canonical completions, exhibit properties like subfitness, regularity, and Booleanness, with implications for topology and modal logic.

Contribution

It provides new characterizations of when related completions are subfit, regular, or Boolean, extending Janowitz's criteria to broader classes of completions.

Findings

Characterizations of subfit, regular, and Boolean completions.

Connections between completion properties and applications in topology and modal logic.

Analysis of stronger distributivity conditions in related completions.

Abstract

Completions play an important r\^ole 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.