On Unitary Hypergroups over the Group

Abstract

A concept of the unitary hypergroup M_H over the group is introduced. If for a given set M and a group H all unitary hypergroups M_H over the group, up to isomorphism, are known, a method is proposed to construct all hypergroups M_H over the group, up to isomorphism. It reduces the problem of description of all hypergroups over the group, up to isomorphism, to the problem of description of all unitary hypergroups over the group, up to isomorphism, and can have many other applications. When M and H are finite, a formula is obtained, which connects the numbers of elements in the classes of isomorphism of all hypergroups over the group and of all unitary hypergroups over the group.