On the hypergroups with four proper pairs and two or three non-scalar elements
For a hypergroup \(H\) denote by \(P(H)=\{(x,y)\in H^2\mid|xy|\geq 2\}\), the set of all proper pairs of \(H\), and by \(S(H)=\{x\in H\mid|xy|=|yx|=1\), \(\forall y\in H\}\), the set of all scalars of \(H\). The authors determine (up to isomorphism) the hypergroups with four proper pairs which also verify the supplementary condition \(\text{Card}(S(H))\in\{2,3\}\). The problem of determination of all hypergroups \(H\) for which \(P(H)\) contains exactly four elements is difficult and for solving it it is necessary to develop long and laborious calculations. The authors surpass these difficulties in an elegant manner and finally obtain a classification of this type of hypergroups. The big variety of examples and the main result given in this paper can be considered as important tools for classifying all finite hypergroups.
scientific article; zbMATH DE number 1036016