Dominions in quasivarieties of universal algebras
type: Item
instance of: scholarly article
title: Dominions in quasivarieties of universal algebras
zbMATH Open document ID: 1066.08006
published in: Studia Logica
publication date: 2005-04-07T00:00:00Z
review text: Let \(\mathcal M\) be a class of algebras, and let \(H\) be a subalgebra of the algebra \(A\), then the dominion of \(H\) in \(A\) (in a class \(\mathcal M\)) is defined by: \[ \text{dom}^{\mathcal M}_A(H)=\{a\in A\mid \forall M\in {\mathcal M}, \forall f,g : A\to M (f| _h=g| _g \Rightarrow a^f=a^g)\}. \] The dominion is intimately connected with the free product of two copies of \(A\) with \(H\) amalgamated, for it is the preimage in \(A\) of the intersection of the two images of \(A\) in the free product. Another useful property of the dominion is that if a quasivariety \(\mathcal M\) is generated by the set \(M\), then \(\text{ dom}^{\mathcal M}_A(H)= \text{dom}^M_A(H)\), provided certain finiteness conditions hold. Also, if \(\mathcal M\) is a quasivariety, \(A\) an arbitary algebra, and \(H\) a finitely generated subalgebra of \(A\), then the set \(L(A,H,{\mathcal M})\) of dominions \(\{\text{dom}^{\mathcal N}_A(H)\}\), where \(\mathcal N\) is a subquasivariety of \(\mathcal M\), is a lattice. The paper ends with an investigation, using examples from group theory, as to whether any of the finiteness conditions in the last two results can be omitted. Three open problems are given.
reviewed by: Sheila Oates-Williams
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08B25
zbMATH DE Number: 2153438
zbMATH Keywords: quasivariety
zbMATH Keywords: dominion
zbMATH Keywords: universal algebra
zbMATH Keywords: group
zbMATH Keywords: lattice
zbMATH Keywords: free amalgamated product
zbMATH Keywords: amalgam
author: Alexander Ivanovich Budkin
MaRDI profile type: Publication
full work available at URL: s11225-005-7127-1
OpenAlex ID: W2092236955
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DOI: 10.1007/S11225-005-7127-1
description: scientific article; zbMATH DE number 2153438