MacNeille completions and canonical extensions

Gehrke, Mai; Harding, John; Venema, Yde

doi:10.1090/S0002-9947-05-03816-X

MacNeille completions and canonical extensions
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by Mai Gehrke, John Harding and Yde Venema PDF

Trans. Amer. Math. Soc. 358 (2006), 573-590 Request permission

Abstract:

Let $V$ be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if $V$ is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety $V$ is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.

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