Bounded distributive lattice expansions

Abstract

A new notion of a canonical extension $\mathbf{A}^{\sigma }$ is introduced that applies to arbitrary bounded distributive lattice expansions (DLEs) $\mathbf{A}$. The new definition agrees with the earlier ones whenever they apply. In particular, for a bounded distributive lattice $\mathbf{A}, \mathbf{A}^{\sigma }$ 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 $f^{\sigma }:\mathbf{A}^{\sigma }\rightarrow \mathbf{B}^{\sigma }$ of an arbitrary map $f:\mathbf{A}\rightarrow \mathbf{B}$ between DLs, and hence to define the canonical extension $\mathbf{A}^{\sigma }$ of an arbitrary DLE $\mathbf{A}$. Together the topologies form a powerful tool for showing that many properties of DLEs are preserved by canonical extensions.