Comparison semigroups and algebras of transformations.
type: Item
instance of: scholarly article
title: Comparison semigroups and algebras of transformations.
zbMATH Open document ID: 1207.20060
published in: Semigroup Forum
publication date: 2010-10-27T00:00:00Z
review text: A set equipped with a certain quaternary comparison operation \((-,-)[-,-]\) is called a `comparison algebra'. A semigroup \((A,\cdot)\) equipped with a comparison operation \((-,-)[-,-]\) which satisfies the equalities \((a,b)[c,d]\cdot e=(a,b)[ce,de]\) and \(e\cdot(a,b)[c,d]=(ea,eb)[ec,ed]\) for all \(a,b,c,d,e\in A\), is called a `comparison semigroup'. A comparison semigroup embeddable in one of the form \(T_X\) (the semigroup of transformations on a set \(X\)) is called `functional'. It is proved that the abstract class of functional comparison semigroups (monoids) is the class of comparison semigroups (resp. monoids). Moreover, using a generalised comparison operation, the latter result is generalised to subalgebras of \(P_X\) (partial transformations on a set \(X\)). Finally, the author expresses a number of operations on partial transformations in the language of comparison semigroups with zero.
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 08A55
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 5806760
zbMATH Keywords: comparison algebras
zbMATH Keywords: comparison operations
zbMATH Keywords: transformation semigroups
zbMATH Keywords: partial transformations
zbMATH Keywords: functional comparison semigroups
author: T. E. Stokes
MaRDI profile type: Publication
full work available at URL: s00233-010-9226-1
OpenAlex ID: W2010477315
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DOI: 10.1007/S00233-010-9226-1
description: scientific article; zbMATH DE number 5806760