Automorphism groups of centralizers of idempotents.
type: Item
instance of: scholarly article
title: Automorphism groups of centralizers of idempotents.
zbMATH Open document ID: 1037.20062
DOI: 10.1016/S0021-8693(03)00499-X
author: João Araújo
author: Janusz Konieczny
published in: Journal of Algebra
publication date: 2003-11-19T00:00:00Z
review text: For a set \(X\), an equivalence relation \(\rho\) on \(X\), and a cross-section \(R\) of the partition \(X/\rho\), the subsemigroup \(T(X,\rho,R)\) of the semigroup \(T(X)\) of full transformations on \(X\), given by \[ T(X,\rho,R)=\{a\in T(X):Ra\subseteq R\text{ and }x\rho y\implies xa\rho ya\}, \] is the centralizer of the idempotent transformation with kernel \(\rho\) and image \(R\). The authors prove that the automorphisms of \(T(X,\rho,R)\) are the inner automorphisms induced by the units of \(T(X,\rho,R)\) and that the automorphism group of \(T(X,\rho,R)\) is isomorphic to the group of units of \(T(X,\rho,R)\). This result generalizes the results of \textit{J. Schreier} [Fundam. Math. 28, 261-264 (1936; Zbl 0016.29503)] and \textit{A. I. Mal'cev} [Mat. Sb., N. Ser. 31(73), 136-151 (1952; Zbl 0048.25204); translation in Am. Math. Soc., Translat., II. Ser. 113, 235-250 (1979)] on the automorphism group of \(T(X)\), and of \textit{Eh. G. Shutov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1961, No. 3(22), 177-184 (1961; Zbl 0101.01405)] and \textit{K. D. Magill}, jun. [J. Aust. Math. Soc. 7, 81-94, 95-107 (1967; Zbl 0148.01502)] on the automorphism group of \(PT(X)\).
reviewed by: Huang Wenxue
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 2005907
zbMATH Keywords: automorphism groups
zbMATH Keywords: centralizers of idempotents
zbMATH Keywords: characteristic subsets
zbMATH Keywords: cross-sections
zbMATH Keywords: inner automorphisms
zbMATH Keywords: transformation semigroups
zbMATH Keywords: groups of units
MaRDI profile type: Publication
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full work available at URL: s0021-8693(03)00499-x
OpenAlex ID: W1975836735
description: scientific article; zbMATH DE number 2005907