New aspects in the theory of complete hypergroups

An algebraic hypercompositional structure is a non-empty set H endowed with at least one hyperoperation, i.e., a multivalued function that associates to any pair of elements x, y in H their hyperproduct x o y representing a subset of H. 1934 is the year when the first hypercompositional structure was introduced. In fact, F. Marty defined the hypergroup as a pair (H, o) formed by a non-empty set H and a multivalued operation (hyperoperation) on the cartesian product H x H with values in the set of non-empty subsets of H, satisfying two axioms: i) the associativity, (x o y) o z=x o (y o z), for any x, y, z in H and ii) the reproductivity, x o H=H= H o x, for any x in H. The aim of this presentation is to recall and gather the main properties of one class of hypergroups, with similar behavior as groups, that attracted (and still attracts) the attention of many researchers during the last 50 years. These are the complete hypergroups, defined in 1970 by Koskas using the notion of complete part. Briefly, a complete part A is a subset of a hypergroup H containing all the hyperproducts of the elements in H having non-empty intersection with A and a hypergroup (H, o) is called complete if the complete closure C(x o y) of any hyperproduct x o y, i.e., the intersection of all complete parts of H containing x o y, is exactly x o y. A very useful characterization of the complete hypergroups was provided later on by P. Corsini, that emphasizes the strong connections that exists between hypergroups and groups. He proved that any complete hypergroup (H, o) may be represented as a union of its non-empty subsets A_g, with g in G, where i) (G, +) is a group; ii) the family {A_g / g in G} is a partition of H; iii) the hyperoperation on H is defined as: if (x,y) in A_g x A_h, with g, h in G, then x o y=A_g+h. The presentation will focus on several properties related to: i) the core (or the heart) of a complete hypergroup; ii) the reversibility and regularity properties; iii) the class equation; iv) the reducibility and fuzzy reducibility properties; v) the grade fuzzy set; vi) the commutativity degree; vii) the Euler's totient function defined on complete hypergroups. These are just some of the topics related to complete hypergroups developed in the last period, that for sure will open new lines of research on hypergroups.