Dedekind-MacNeille and Related Completions: Subfitness, Regularity, and Booleanness

Authors

G. Bezhanishvili, New Mexico State UniversityFollow
F. Dashiell Jr., Chapman UniversityFollow
M. A. Moshier, Chapman UniversityFollow
Joanne Walters-Wayland, Chapman UniversityFollow

Document Type

Article

Publication Date

4-9-2025

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.

Comments

This article was originally published in Order in 2025. https://doi.org/10.1007/s11083-024-09691-9

Recommended Citation

Bezhanishvili, G., Dashiell Jr, F., Moshier, M.A. et al. Dedekind-MacNeille and Related Completions: Subfitness, Regularity, and Booleanness. Order (2025). https://doi.org/10.1007/s11083-024-09691-9

Peer Reviewed

1

Copyright

The authors

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.