Canonical Extensions and Discrete Dualities for Finitely Generated Varieties of Lattice-based Algebras

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety \({\mathcal {A}}\) is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show that topological and discrete dualities for \({\mathcal {A}}\) exist in partnership.