Bounded distributive lattice expansions

Abstract

A new notion of a canonical extension Aσ is introduced that applies to arbitrary bounded distributive lattice expansions (DLEs) A. The new definition agrees with the earlier ones whenever they apply. In particular, for a bounded distributive lattice A, Aσ has the same meaning as before. A novel feature is the introduction of several topologies on the universe of the canonical extension of a DL. One of these topologies is used to define the canonical extension fσ: Aσ → Bσ of an arbitrary map f: A → B between DLs, and hence to define the canonical extension Aσ of an arbitrary DLE A. Together the topologies form a powerful tool for showing that many properties of DLEs are preserved by canonical extensions.