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A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras. (English) Zbl 1231.06010

Summary: This paper investigates completions in the context of finitely generated lattice-based varieties of algebras. It is shown that, for such a variety \(\mathcal A\), the order-theoretic conditions of density and compactness, which characterise the canonical extension of (the lattice reduct of) any \(A\in \mathcal A\), 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 \(n_{\mathcal A}(A)\) whose underlying algebra belongs to \(\mathcal A\). Furthermore, each of the operations of \(n_{\mathcal A}(A)\) coincides with both the \(\sigma \)-extension and the \(\pi \)-extension of the corresponding operation on \(A\), with which a canonical extension is customarily equipped. Thus, in particular, the variety \(\mathcal A\) 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.

MSC:

06B23 Complete lattices, completions
06B20 Varieties of lattices
06B30 Topological lattices
06F25 Ordered rings, algebras, modules
08B25 Products, amalgamated products, and other kinds of limits and colimits
22A30 Other topological algebraic systems and their representations
54H12 Topological lattices, etc. (topological aspects)
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