Hypergroups of type \(U\) and homology of complexes
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.
scientific article; zbMATH DE number 840231
scientific article