On automorphisms of transformation semigroups
type: Item
instance of: scholarly article
title: On automorphisms of transformation semigroups
zbMATH Open document ID: 0793.20060
DOI: 10.1007/BF02573654
author: Inessa Levi
author: Graham R. Wood
published in: Semigroup Forum
publication date: 1994-08-14T00:00:00Z
full work available at URL: https://eudml.org/doc/135297
review text: Let \(p\) and \(q\) be infinite cardinals where \(p \geq q\) and let \(X\) be a set with \(| X| \geq p\). Let \({\mathcal E} = \{{\mathcal E}_ i: i \in I\}\) be a family of distinct equivalence relations on \(X\) such that \(| X /{\mathcal E}_ i| = p\) for all \(i\in I\). A subset \(A\) of \(X\) is well separated (ws) by \(\mathcal E\) if \(| A| = p\) and \({\mathcal E}_ i \cap (A \times A)\) is the identity relation on \(A\) for all \(i \in I\). Let \({\mathcal C}_ t = \{A: A \text{ is ws and }A \subseteq B\) and \(| B\setminus A|= t\) for some ws \(B\)\} where \(q \leq t\leq p\). In the event \(X\) contains a ws set, the semigroup \(CT(X,{\mathcal E},p,q)\) consisting of all selfmaps \(f\) of \(X\) with \(\pi(f) \in {\mathcal E}\) and \(R(f) \in {\mathcal C}_ q\) is the well known Croisot-Tessier semigroup on \(X\), \(\mathcal E\) of type \((p,q)\) where \(\pi(f)\) is the decomposition induced by \(f\) and \(R(f)\) is the range of \(f\). If \(\mathcal B\) is an equivalence relation on \(X\), the expression \([x] \in {\mathcal B}\) means that \([x]\) is the equivalence class induced by \(\mathcal B\) which contains the element \(x\). The authors generalize the notion of a Croisot-Tessier semigroup as follows. Let \(r\) and \(p\) be infinite cardinals such that \(r \leq p\) and let \(\mathcal A\) be an equivalence relation on \(X\). Denote by \({\mathcal A}^{(r)}\) the family of all equivalence relations \(\mathcal B\) on \(X\) such that \({\mathcal A} \subseteq {\mathcal B}\) and \(| [x]/{\mathcal A}| < r\). The family \({\mathcal A}^{(r)}\) is referred to as the family of \(r\) glueings of \(\mathcal A\). Now let \({\mathcal E}^{(r)} = \bigcup \{{\mathcal E}^{(r)}_ i: i\in I\}\) be the family of \(r\) gluings of \(\mathcal E\) and let \(S\) denote the semigroup consisting of all those selfmaps \(f\) of \(X\) such that \(R(f) \in {\mathcal C}_ q\) and \(\pi(f) \in {\mathcal E}^{(r)}\). Next, let \({\mathcal E}^ \# = \{{\mathcal A} \in {\mathcal E}^{(r)}: \pi(t) = \…
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 492606
zbMATH Keywords: equivalence relations
zbMATH Keywords: well separated
zbMATH Keywords: automorphisms
zbMATH Keywords: Croisot-Tessier semigroups
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
cites work: Q5564342
cites work: Automorphisms of Croisot-Teissier semigroups
cites work: Green's relations on Croisot-Teissier semigroups
cites work: Range sets and partition sets in connection with congruences and algebraic invariants
cites work: Q3208900
cites work: Regular congruences on Croisot-Teissier and Baer-Levi semigroups
cites work: Q4767916
cites work: Completely simple congruences on Croisot-Teissier semigroups
OpenAlex ID: W2046217129
description: scientific article; zbMATH DE number 492606