Links Between HX-Groups and Hypergroups

Abstract

The concept of an [Formula: see text]-group is an upgrade of the concept of a group, in which a new operation is defined on the family of non-empty subsets of a group. If this new support set together with the new operation is a group, then we call it an [Formula: see text]-group. On the other hand, a hyperoperation is a mapping having the same codomain as the operation of an [Formula: see text]-group, i.e., the family of non-empty subsets of the initial set, but a different domain — the set itself. This could be (and was indeed) a source of confusion, which is clarified in this paper. Moreover, [Formula: see text]-groups naturally lead to constructions of hypergroups. The links between these two algebraic concepts are presented, with the aim of reviving the old notion of an [Formula: see text]-group in the current research on algebraic hyperstructures. One of such existing links and one newly established link are also discussed.