Summary: There are two natural ways to extend an arbitrary map between (the carriers of) two lattices to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions and for which kind of maps the two extensions coincide.
Our perspective involves a number of topologies on lattice completions, including the Scott topologies and topologies that are induced by the original lattice. We provide a characterization of the MacNeille completion in terms of these induced topologies.
We then turn to expansions of lattices with additional operations, and address the question of which equational properties of such lattice expansions are preserved under various types of MacNeille completions that can be defined for these algebras. For a number of cases, including modal algebras and residuated (ortho)lattice expansions, we provide reasonably sharp sufficient conditions on the syntactic shape of equations that guarantee preservation. Generally, our results show that the more residuation properties the primitive operations satisfy, the more equations are preserved.