G. Bezhanishvili, F. Dashiell Jr, A. Moshier, J. Walters-Wayland
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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.
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