A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras

This paper investigates completions in the context of finitely generated lattice-based varieties of algebras. It is shown that, for such a variety A, the order-theoretic conditions of density and compactness which characterise the canonical extension of (the lattice reduct of) any A∈A have truly topological interpretations. In addition, a particular realisation is presented of the canonical extension of A; this has the structure of a topological algebra nA(A) whose underlying algebra belongs to A. Furthermore, each of the operations of nA(A) coincides with both the σ-extension and the π-extension of the corresponding operation on A, with which a canonical extension is customarily equipped. Thus, in particular, the variety A is canonical, and all its operations are smooth. The methods employed rely solely on elementary order-theoretic and topological arguments, and by-pass the subtle theory of canonical extensions that has been developed for lattice-based algebras in general.