On coconnected algebras
type: Item
instance of: scholarly article
title: On coconnected algebras
zbMATH Open document ID: 1026.08003
DOI: 10.1016/S0022-4049(02)00245-1
author: Věra Trnková
author: Jiří Sichler
published in: Journal of Pure and Applied Algebra
publication date: 2003-04-09T00:00:00Z
review text: Let \(A\) be an object in a category \({\mathcal A}\) with finite products and let \(n\geq 2\) be finite. For the \(n\)-fold power \(A^n\) of \(A\), let \(\{\pi^{(n)}_i; i\in n\}\) be the collection of its product projections. \(A\) is said to be \(n\)-coconnected if every \({\mathcal A}\)-morphism \(f: A^n\to A\) has a decomposition \(f= h\circ\pi^{(n)}_i\) for some \(h\in \text{End }A\) and \(i\in n= \{0,\dots, n-1\}\). 2-coconnected objects are called coconnected. \(A\) is said to be fully coconnected if \(A\) is \(n\)-coconnected for every \(n\). In the paper it is shown that coconnectedness presents no obstacle to monoid representation by endomorphisms of algebras from certain universal varieties. It is proved that for every monoid \(M\) there exist arbitrarily large \((0,1)\)-lattices \(A_0\), \(A_1\) with \(\text{End }A_0\cong \text{End }A_1\cong M\) such that \(A_0\) is not coconnected while \(A_1\) is fully coconnected, and that there is an arithmetical variety with this property. Further, the authors show that the variety \(\text{Alg}(1,1)\) of all algebras with two unary operations is also of this kind, and that coconnected monounary algebras are determined by their endomorphism monoids. A partial answer to the problem of whether every 3-coconnected algebra is fully coconnected is given.
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 08A35
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 08A60
zbMATH DE Number: 1895981
zbMATH Keywords: coconnected algebra
zbMATH Keywords: universal class
zbMATH Keywords: endomorphism monoid
zbMATH Keywords: monoid representation
zbMATH Keywords: universal varieties
MaRDI profile type: Publication
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full work available at URL: s0022-4049(02)00245-1
OpenAlex ID: W2065349883
description: scientific article; zbMATH DE number 1895981