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

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 situated between the two 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 are useful in pointfree topology since (unlike the Dedekind-MacNeille completion) they satisfy stronger forms of distributivity.