Hypergroups of type \(C\)
type: Item
instance of: scholarly article
title: Hypergroups of type \(C\)
zbMATH Open document ID: 0771.20025
DOI: 10.1007/BF02845077
author: Yves Sureau
publication date: 1992-09-27T00:00:00Z
review text: A hypergroup \((H,\circ)\) verifying the following two conditions: i) there exists \(e\in H\) such that \(x\circ e=x\), for any \(x\in H\) (\(e\) is the (right) identity of \(H\)); ii) for any \(x,y,z\) in \(H\), \(x\circ y\cap x\circ z\neq\emptyset\) implies \(e\circ y=e\circ z\); is said to be a hypergroup of type \(C\) (at the right) ((right) weak cogroup). If moreover \((H,\circ)\) verifies the condition: iii) \(\text{Card}(x\circ y)=\text{Card}(e\circ y)\), for any \(x,y\) in \(H\); then \((H,\circ)\) is called a (right) cogroup. Any \(D\)-hypergroup, \((G/{\mathcal G},\circ)\), where \(G\) is a group and \(\mathcal G\) is a (not necessarily normal) subgroup of \(G\), is a cogroup[\textit{J. Eaton}, Duke Math. J. 6, 101-107 (1940; Zbl 0027.00804)] but there exist cogroups which are not \(D\)-hypergroups[\textit{Y. Utumi}, Osaka Math. J. 1, 73-80 (1949; Zbl 0036.293)]. In this paper the most important properties of the hypergroups of type \(C\) and many examples which situate these hypergroups among the classes of hypergroups already known are presented. The morphisms and the quotient hypergroups of the hypergroups of type \(C\) are studied. Among the main results established in the paper, the following theorem presents a construction which is analogous to the semidirect product of groups: Theorem. Let \((H,\circ)\) be a cogroup and let \(e\) be the right identity of \(H\). Consider \(\mathcal G\) a group of permutations of \(H\) which let invariant the partition \(P=\{e\circ x\mid x\in H\}\) of \(H\) and \((\phi_ x)_{x\in H}\) a family of surjective maps \(\phi_ x: H\to H\), for any \(x\in H\). Assume that: i) \(\phi_ x(e\circ y)=e\circ y\), for any \(x,y\) in \(H\); ii) \(\phi_ e=1_ H\); iii) \(\phi_ x(\sigma(\phi_ y(\tau(z))))=\phi_{\phi_ x(\sigma(y))}((\sigma\circ\tau)(z))\), for any \(\sigma,\tau\) in \(\mathcal G\) and \(x,y,z\) in \(H\); Define on \(G=H\times {\mathcal G}\) the operation ``\(\cdot\)'' by: \((x,\sigma)\cdot (y,\tau)=(\phi_ x(\sigma(y)),\si…
Mathematics Subject Classification ID: 20N20
zbMATH DE Number: 60499
zbMATH Keywords: hypergroups of type \(C\)
zbMATH Keywords: morphisms
zbMATH Keywords: quotient hypergroups
zbMATH Keywords: semidirect product
zbMATH Keywords: right identity
zbMATH Keywords: \(D\)-hypergroups
zbMATH Keywords: weak cogroups
reviewed by: Marin Guţan
MaRDI profile type: Publication
cites work: Polygroups derived from cogroups
cites work: Q3791372
cites work: Theory of cogroups
cites work: Les cogroupes et la construction de Utumi. (Cogroups and the Utumi construction)
cites work: Groupoides, demi-hypergroupes et hypergroupes
cites work: Q5779934
cites work: Q5794684
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 60499