Natural extensions and profinite completions of algebras
type: Item
instance of: scholarly article
title: Natural extensions and profinite completions of algebras
zbMATH Open document ID: 1232.08006
published in: Algebra Universalis
publication date: 2011-12-13T00:00:00Z
review text: A profinite completion of a given algebra \(A\) is defined to be the inverse limit algebra formed from finite quotient algebras of \(A\) by congruences of finite index, where the indexing set is the set of congruences of finite index and ordering is reverse inclusion. The paper investigates profinite completions of residually finite algebras drawing on ideas of natural dualities. Given a class \(V=\mathrm{ISP}(M)\), where \(M\) is a set of finite algebras, it is shown that each \(A \in V\) embeds as a topologically dense subalgebra of a topological algebra which is a natural extension of \(A\) and this algebra is isomorphic, topologically and algebraically, to the profinite completion of \(A\). It is shown how the natural extensions can be described as families of relation-preserving maps. For an algebra of a finitely generated variety of lattice-based algebras, the natural extension provides a realization of the canonical extension. The paper concludes with an exhaustive survey of classes of algebras to which the main theorem do, or do not, apply.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 06D50
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08C20
Mathematics Subject Classification ID: 18A30
zbMATH DE Number: 5988045
zbMATH Keywords: profinite completion
zbMATH Keywords: residually finite
zbMATH Keywords: natural extension
zbMATH Keywords: canonical extension
author: Brian A. Davey
author: Maria João Gouveia
author: Miroslav Haviar
author: Hilary A. Priestley
MaRDI profile type: Publication
full work available at URL: s00012-011-0155-y
OpenAlex ID: W1998618342
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Natural extensions and profinite completions of algebras (DOI)
DOI: 10.1007/S00012-011-0155-Y