Equivalences, \(\alpha\)-semigroups and \(\alpha\)-congruences
type: Item
instance of: scholarly article
title: Equivalences, \(\alpha\)-semigroups and \(\alpha\)-congruences
zbMATH Open document ID: 0804.20046
DOI: 10.1007/BF02573470
author: Huisheng Pei
published in: Semigroup Forum
publication date: 1995-01-17T00:00:00Z
full work available at URL: https://eudml.org/doc/135333
review text: Let \(X\) be a set with more than one point. An \(\alpha\)-semigroup, \(T(X)\), is any semigroup of selfmaps of \(X\) which contains the identity map and all constant maps. An equivalence relation \(E\) on \(X\) is a \(T\)- equivalence relation if \((f(x), f (y)) \in E\) for all \((x, y) \in E\). The complete lattice of all \(T\)-equivalences on \(X\) is denoted by \(T\text{eq}(X)\) and the complete lattice of all equivalence relations on \(X\) is denoted by \({\mathcal E}(X)\). The \(T\)-equivalences turned out to be useful in the search for the smallest proper congruence on \(T(X)\). Specifically, it was shown by \textit{K. H. Hofmann} and \textit{K. D. Magill} jun. [Glasg. Math. J. 30, 301-313 (1988; Zbl 0681.54018)] that \(T(X)\) has a smallest proper congruence if and only if there is a smallest proper \(T\)-equivalence and moreover, one can describe the smallest proper congruence in terms of that \(T\)-equivalence. The work in this paper was motivated by the following question. Given a complete sublattice \(L\) of \({\mathcal E}(X)\) with \(\Delta(X),X \times X\in L\), where \(\Delta(X)\) is the diagonal of \(X\), does there exist an \(\alpha\)-semigroup \(T(X)\) such that \(T\text{eq}(X) = L\)? The question is left unanswered but to get some idea of the type of result obtained, choose \(E \in {\mathcal E}(X)\) and define \(T_ E(X) = \{f \in {\mathcal T}_ X : (f(a), f(b))\in E\;\forall(a, b) \in E\}\) where \({\mathcal T}_ X\) is the semigroup of all selfmaps of \(X\). The author proves, for example, that if \(E,F \in{\mathcal E}(X)\) are noncomparable and if \(E \cap F \neq \Delta(X)\) and \(E \vee F \neq X\times X\), then \(T\text{eq}(X) = \{\Delta(X), (E \cap F), E, F, (E \vee F), X\times X\}\) where \(T(X) = T_ E(X) \cap T_ F(X)\). He goes on to conjecture that if \(L\) is an arbitrary sublattice of \({\mathcal E}(X)\) containing both \(\Delta(X)\) and \(X \times X\), then \(T(X) = \bigcap\{T_ E(X) : E \in L\}\) is the greatest \(\alpha\)-semigroup with \(…
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 611114
zbMATH Keywords: lattice of equivalence relations
zbMATH Keywords: \(\alpha\)-semigroup
zbMATH Keywords: semigroup of selfmaps
zbMATH Keywords: \(T\)-equivalence relation
zbMATH Keywords: \(T\)-equivalences
zbMATH Keywords: smallest proper congruence
zbMATH Keywords: \(\alpha\)-congruences
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
cites work: The smallest proper congruence on S(X)
cites work: Q4126563
cites work: Some Homomorphism Theorems for a Class of Semigroups
cites work: The \(\alpha\)-congruences on \(S(X)\) and the \(S\)-equivalences on \(X\)
description: scientific article; zbMATH DE number 611114