From HX-Groups to HX-Polygroups

HX-groups are a natural generalization of groups that are similar in construction to hypergroups. However, they do not have to be considered as hypercompositional structures like hypergroups; instead, they are classical groups. After clarifying this difference between the two algebraic structures, we review the main properties of HX-groups, focusing on the regularity property. An HX-group G on a group G with the identity e is called regular whenever the identity E of G contains e. Any regular HX-group may be characterized as a group of cosets, and equivalent conditions for describing this property are established. New properties of HX-groups are discussed and illustrated by examples. These properties are uniformity and essentiality. In the second part of the paper, we introduce a new algebraic structure, that of HX-polygroups on a polygroup. Similarly to HX-groups, we propose some characterizations of HX-polygroups as polygroups of cosets or double cosets. We conclude the paper by proposing several lines of research related to HX-groups.