Let V 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 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 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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CITATION STYLE
Gehrke, M., Harding, J., & Venema, Y. (2005). MacNeille completions and canonical extensions. Transactions of the American Mathematical Society, 358(2), 573–590. https://doi.org/10.1090/s0002-9947-05-03816-x