Abstract :
The study of various kinds of algebraic hypergroups is unified in the theory of transposition hypergroups. The hyperoperation in any hypergroup has two inverses given bya/b = {x a set membership, variant xb} andb/a = {x a set membership, variant bx}. A transposition hypergroup is a hypergroup where transposition,b/a ∩ c/d ≠ empty set impliesad ∩ bc ≠ empty set, holds. The algebra of transposition hypergroups is developed. Closed subhypergroupsNthat are reflexive,a/N = N/a, have distinguished structural significance. The quotient space of a transposition hypergroup modulo a reflexive closed subhypergroup forms a transposition hypergroup that is a polygroup. Then generalizations of the isomorphism theorems and the Jordan–Hölder theorem of group theory are obtained.
