
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.
Structure and representation theory of distributive lattices, canonical extension algebras, extension of bounded distributive lattices, topologies
Structure and representation theory of distributive lattices, canonical extension algebras, extension of bounded distributive lattices, topologies
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