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MacNeille completions and canonical extensions. (English) Zbl 1083.06009
Summary: 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.

MSC:
06B23 Complete lattices, completions
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
06B20 Varieties of lattices
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