Transactions of the American Mathematical Society

Author(s): Mai Gehrke; John Harding; Yde Venema
Journal: Trans. Amer. Math. Soc. 358 (2006), 573-590.
MSC (2000): Primary 06B23, 03G10; Secondary 03B45, 03C05, 03G25, 06E25
Posted: June 21, 2005
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Abstract: Let

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

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

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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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 06B23, 03G10, 03B45, 03C05, 03G25, 06E25

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: 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
Posted: 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.
Copyright of article: Copyright 2005, American Mathematical Society

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