Residually small varieties without rank
type: Item
instance of: scholarly article
title: Residually small varieties without rank
zbMATH Open document ID: 1058.08008
author: Hans-E. Porst
published in: Applied Categorical Structures
publication date: 2005-01-17T00:00:00Z
review text: We recall that a variety \(\mathbb V\) is residually small if any class of non-isomorphic subdirectly irreducible algebras from \(\mathbb V\) is a set. For a set functor \(F\), let \(\text{COAL}(F)\) denote the category of all \(F\)-coalgebras and their homomorphisms and let \(\mathbb V_F\) be a variety of (unranked) algebras equivalent to the dual of \(\text{COAL}(F)\). It is proved that the variety of complete atomic Boolean algebras and the variety of commutative \(C^{*}\)-unital algebras are residually small and Birkhoff's subdirect representation theorem is true for these varieties. On the other hand, the variety of compact Hausdorff spaces is residually small but Birkhoff's subdirect decomposition theorem does not hold. These observations are generalized. If \(F\) is a set-accessible functor preserving intersections then Birkhoff's subdirect decomposition theorem holds in \(\mathbb V_F\) and \(\mathbb V_F\) is residually small.
reviewed by: Václav Koubek
Mathematics Subject Classification ID: 08B26
Mathematics Subject Classification ID: 18C15
zbMATH DE Number: 2128569
zbMATH Keywords: subdirect representation
zbMATH Keywords: subdirectly irreducible algebra
zbMATH Keywords: residually small variety
zbMATH Keywords: complete atomic Boolean algebra
MaRDI profile type: Publication
full work available at URL: b:apcs.0000040385.40141.da
OpenAlex ID: W1964788647
DOI: 10.1023/B:APCS.0000040385.40141.DA
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description: scientific article; zbMATH DE number 2128569