Hereditary radicals and 0-bands of semigroups
type: Item
instance of: scholarly article
title: Hereditary radicals and 0-bands of semigroups
zbMATH Open document ID: 0661.20043
DOI: 10.1007/BF02573218
published in: Semigroup Forum
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134941
review text: With any abstract class \({\mathcal R}\) of semigroups is associated the map \(S\to \rho (S)\), the union of the ideals of S which belong to \({\mathcal R}\). This map is a ``radical'' (and \({\mathcal R}\) is a ``radical class'') if (i) \({\mathcal R}\) is closed under Rees quotients, (ii) \(S\in {\mathcal R}\) if and only if \(\rho (S)=S\) and (iii) \(\rho (S/\rho (S))=\{0\}\) for any S. For example, the class \({\mathcal N}\) of nilsemigroups determines the ``Clifford'' radical and the class \({\mathcal L}\) of locally nilpotent semigroups determines the ``Shevrin'' radical. The author studies the interaction between these, and other, radicals, such as the ``McCoy'' radical, with 0-band decompositions of semigroups. A radical \(\rho\) is ``restorable by the components of every 0-band'' if whenever S is a 0-band \(\{S_{\alpha}:\) \(\alpha\in \Omega \}\), (where the components \(S_{\alpha}\) are 0-disjoint), then \(\rho\) (S) is the largest ideal contained in \(\cup \{\rho (S_{\alpha}):\) \(\alpha\in \Omega \}\). Amongst other results, it is shown that the two radicals above have this property.
Mathematics Subject Classification ID: 20M11
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 4079760
zbMATH Keywords: ideals
zbMATH Keywords: Rees quotients
zbMATH Keywords: nilsemigroups
zbMATH Keywords: ``Clifford'' radical
zbMATH Keywords: locally nilpotent semigroups
zbMATH Keywords: ``Shevrin'' radical
zbMATH Keywords: 0-band decompositions
zbMATH Keywords: radicals
author: A. V. Kelarev
reviewed by: Peter R. Jones
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4079760