Canonical completions and MacNeille completions
by
John Harding
New Mexico State University
Coauthors: Mai Gehrke and Yde Venema

Two common methods to complete a Boolean algebra are the MacNeille completion and the canonical completion. The MacNeille completion is the familiar generalization of Dedekind's completion by cuts. The canonical completion is obtained by embedding a Boolean algebra into the power set of its Stone space. For a Boolean algebra with additional operations, such as a modal algebra, one is often interested in whether these operations can be lifted to the MacNeille, or canonical, completion in a manner that preserves some desired set of identities.

Clearly the MacNeille completion can be considered in the wider context. We consider here a generalization of the canonical completion to bounded lattices with additional operations. We discuss circumstances under which a variety of bounded lattices with additional operations is closed under MacNeille, or canonical, completions. Further, under fairly modest restrictions, we show that any variety of bounded lattices with additional operations that is closed under MacNeille completions is also closed under canonical completions.

Date received: June 3, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoc-02.