The centralizer of an element in a reversible hypergroup
type: Item
instance of: scholarly article
title: The centralizer of an element in a reversible hypergroup
zbMATH Open document ID: 0833.20080
DOI: 10.1007/BF02844252
author: Violeta Leoreanu
publication date: 1996-03-12T00:00:00Z
review text: Let \(H\) be a hypergroup and \(I(H)\) be the set of its identities (an element \(e \in H\) is an identity of \(H\) if \(e \in ex \cap xe\), for every \(x \in H)\). If \(x \in H\), denote by \(i(x) = \{y \in H \mid xy \cap yx \cap I(H) \neq \emptyset\} =\) the set of the inverses of \(x\) and by \({\mathcal C}(x)\) the least complete part of \(H\) containing \(x\). We also consider \(A(x) = \{y \in H \mid {\mathcal C}(xy) \cap {\mathcal C}(yx) \neq \emptyset\}\) and \({\mathcal C}_H(x) = A(x) \cup A^2(x) \cup A^3(x) \cup \dots\). The hypergroup \(H\) is regular if \(I(H) \neq \emptyset\) and \(i(x) \neq \emptyset\), for every \(x \in H\). If, more than that, for every \(x, y, z\) in \(H\) satisfying \(x \in yz\) there exist \(y' \in i(y)\) and \(z' \in i(z)\) such that \(z \in y' x\) and \(y \in xz'\) then \(H\) is reversible. The aim of this paper is to introduce a notion of centralizer for regular and reversible hypergroups and to establish some of its properties. Thus it is shown that, for every \(x \in H\), the set \({\mathcal C}_H(x)\) is a superhypergroup of \(H\), called the centralizer of \(x\) in \(H\). Particularly, for complete hypergroups, whence for groups, too, the centralizer of an element \(x\) is \({\mathcal C}(x) = \{y \in H \mid xy = yx\}\) so it coincides with the classical one. It is also established that in a regular and reversible hypergroup \(H\) the following conditions hold, for every \(x\) in \(H\): (i) \(({\mathcal C}(x) \cup \omega_H) \subset {\mathcal C}_H(x)\); (ii) \({\mathcal C}_H(x) = {\mathcal C}_H(x')\), for every \(x' \in i(x)\); (iii) \({\mathcal C}_H(x) = H\) if \(x \in \omega_H\); (iv) \({\mathcal C}_H (x) = {\mathcal C}_H (y)\), if \(xy \cap \omega_H \neq \emptyset\).
Mathematics Subject Classification ID: 20N20
zbMATH DE Number: 790784
zbMATH Keywords: regular hypergroups
zbMATH Keywords: identities
zbMATH Keywords: inverses
zbMATH Keywords: complete part
zbMATH Keywords: centralizer
zbMATH Keywords: reversible hypergroups
zbMATH Keywords: complete hypergroups
reviewed by: Marin Guţan
MaRDI profile type: Publication
cites work: Q3137919
cites work: Groupoides, demi-hypergroupes et hypergroupes
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 790784