Transposition hypergroups: Noncommutative join spaces
A transposition hyperqroup is a hypergroup \(H\) for which: \((b\setminus a)\cap(c/d)\neq\emptyset\) implies \(ad\cap bc\neq\emptyset\), for every \(a\), \(b\), \(c\), \(d\) in \(H\), where \(b\setminus a=\{x\in H\mid a\in bx\}\) and \(c/d=\{x\in H\mid c\in xd\}\). Introducing this notion, the author treats from a unique point of view several algebraic hyperstructures (join spaces, weak cogroups, double coset spaces, polygroups, canonical hypergroups, groups) instead to study them separately (as usually), all of them being transposition hypergroups. The (closed and reflexive) subhypergroups and the quotient spaces of a transposition hypergroup are studied. The three isomorphism theorems and the theorem of Jordan-Hölder of group theory are derived in the context of transposition hypergroups. The paper is very clear and very well written.
scientific article; zbMATH DE number 1014033