Expanding varieties by monoids of endomorphisms
type: Item
instance of: scholarly article
title: Expanding varieties by monoids of endomorphisms
zbMATH Open document ID: 0599.08011
DOI: 10.1007/BF01194525
published in: Algebra Universalis
publication date: 1983-01-01T00:00:00Z
review text: Let \({\mathcal V}(M)\) denote the variety obtained from a variety \({\mathcal V}\) and a monoid M by adding a unary operation to the language of \({\mathcal V}\) for each element of M and by adding axioms to guarantee that the new functions form a submonoid (isomorphic to M) of the endomorphism monoid of each algebra of \({\mathcal V}\). This paper examines various universal algebraic properties for the expanded variety \({\mathcal V}(M)\). The main results include: (1) \({\mathcal V}\) is equivalent to a subvariety of \({\mathcal V}(M)\); (2) for \({\mathcal V}\) nontrivial, \({\mathcal V}(M)\) is locally finite [resp. a discriminator variety] if and only if \({\mathcal V}\) is locally finite [resp. a discriminator variety] and M is finite [resp. a finite group]; (3) for \({\mathcal V}\) nontrivial, if \({\mathcal V}(M)\) is finitely generated, then \({\mathcal V}\) is finitely generated and M is finite; (4) \({\mathcal V}(M)\) is abelian if and only if \({\mathcal V}\) is abelian. A structure theorem is also obtained when \({\mathcal V}\) is a congruence modular abelian variety. Associated with such a variety are a ring R(\({\mathcal V})\) and a variety of left R(\({\mathcal V})\)-modules. After describing a generating set for R(\({\mathcal V}(M))\), the authors show that R(\({\mathcal V}(M))\) is isomorphic to the ring of all R(\({\mathcal V})\)-valued functions on M which are 0 on all except finitely many arguments.
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 08A35
zbMATH DE Number: 3966165
zbMATH Keywords: endomorphism monoid
zbMATH Keywords: expanded variety
zbMATH Keywords: discriminator variety
zbMATH Keywords: congruence modular abelian variety
author: Stanley N. Burris
author: Matthew Valeriote
MaRDI profile type: Publication
cites work: Q4101811
cites work: Undecidability of the Theory of Abelian Groups with a Subgroup
cites work: The first order theory of Boolean algebras with a distinguished group of automorphisms
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cites work: Sheaf Constructions and Their Elementary Properties
full work available at URL: https://doi.org/10.1007/bf01194525
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description: scientific article; zbMATH DE number 3966165