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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

MacNeille completions and canonical extensions


Authors: Mai Gehrke, John Harding and Yde Venema
Journal: Trans. Amer. Math. Soc. 358 (2006), 573-590
MSC (2000): Primary 06B23, 03G10; Secondary 03B45, 03C05, 03G25, 06E25
Published electronically: June 21, 2005
MathSciNet review: 2177031
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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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Additional Information

Mai Gehrke
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: mgehrke@nmsu.edu

John Harding
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: jharding@nmsu.edu

Yde Venema
Affiliation: Institute for Logic, Language and Computation, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands
Email: yde@science.uva.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03816-X
PII: S 0002-9947(05)03816-X
Keywords: MacNeille completion, canonical extension, lattices, lattice ordered algebras, Boolean algebra with operators
Received by editor(s): January 28, 2004
Published electronically: June 21, 2005
Additional Notes: The authors express their gratitude to the anonymous referee for carefully reading and commenting on the manuscript, and, in particular, for making a valuable suggestion. Thanks are also due to Tadeusz Litak and Rob Goldblatt for comments on earlier versions of this paper. The first author’s research was partially supported by grant NSF01-4-21760 of the USA National Science Foundation.
Article copyright: © Copyright 2005 American Mathematical Society