  A fundamental result from Boolean modal logic states that a first-order
definable class of Kripke frames defines a logic that is validated by all of
its canonical frames. We generalise this to the level of non-distributive
logics that have a relational semantics provided by structures based on
polarities. Such structures have associated complete lattices of stable
subsets, and these have been used to construct canonical extensions of
lattice-based algebras. We study classes of structures that are closed under
ultraproducts and whose stable set lattices have additional operators that are
first-order definable in the underlying structure. We show that such classes
generate varieties of algebras that are closed under canonical extensions. The
proof makes use of a relationship between canonical extensions and MacNeille
completions.
