MacNeille completions and canonical extensions

Abstract

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.