MacNeille completions and canonical extensions
Mai Gehrke, John Harding, Yde Venema

Abstract:
Let $V$ be a variety of monotone bounded lattice expansions, that is,
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.

Keywords: MacNeille completion, canonical extension, lattice ordered
algebra, Boolean algebra with operators