Hypergroups of type \(U\) and homology of complexes
type: Item
instance of: scholarly article
title: Hypergroups of type \(U\) and homology of complexes
zbMATH Open document ID: 0839.20079
DOI: 10.1007/BF01190968
published in: Algebra Universalis
publication date: 1996-06-10T00:00:00Z
review text: A subhypergroup \(h\) of a hypergroup \(H\) is ultraclosed if for every \(x\) in \(H\) the sets \(xh\cap (H\setminus h)\) and \(hx\cap (H\setminus h)\) are empty. If \(f: H\to K\) is a morphism of hypergroups its kernel is defined by \(\text{Ker f}= f^{-1} (U)\), where \(U\) is the intersection of all ultraclosed subhypergroups of \(K\). Thus, in the particular case of groups this notion coincides to the classical one \(\text{Ker } f= f^{-1} (e)\), \(e\) being the neutral element of the group \(K\). In this paper the notion of kernel of a morphism of hypergroups is used in a supercategory of the category of groups, namely the category of the hypergroups of type \(U\) on the right. Recall that a hypergroup \(H\) is of type \(U\) on the right if 1) there exists \(\varepsilon\in H\) such that \(x\varepsilon =x\), for any \(x\in H\); 2) if \(x,y\) are in \(H\) and \(x\in xy\) then \(y= \varepsilon\). Thus a notion of exact sequence is introduced and the existence of Ker-Coker sequences is established. By analogy to the homology of groups, complexes of hypergroups and hypergroups of homology are introduced and their functoriality is proved.
Mathematics Subject Classification ID: 20N20
Mathematics Subject Classification ID: 20J15
Mathematics Subject Classification ID: 18G60
zbMATH DE Number: 840231
zbMATH Keywords: category of hypergroups of type \(U\)
zbMATH Keywords: ultraclosed subhypergroups
zbMATH Keywords: morphism of hypergroups
zbMATH Keywords: category of groups
zbMATH Keywords: Ker-Coker sequences
zbMATH Keywords: complexes of hypergroups
zbMATH Keywords: hypergroups of homology
zbMATH Keywords: functoriality
author: Domenico Freni
author: Yves Sureau
reviewed by: Marin Guţan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 840231