The Class Equation and the Commutativity Degree for Complete Hypergroups

Abstract

The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.

Commutativity and Completeness Degrees of Weakly Complete Hypergroups

“…Indeed, the completeness degree of weakly complete hypergroups admits simple closed formulas. Furthermore, it can be related to the commutativity degree, which has been recently brought into hypercompositional algebra from group theory [13,18].…”

“…Recently, the concept of commutativity degree has been introduced also in hypergroup theory [13,18]. In particular, in [13] the authors defined the commutativity degree of a finite hypergroup (H, •) as…”

“…Hence, a hypergroup (H, •) is complete if and only if x • y = C(x • y), for all x, y ∈ H. Complete hypergroups have been the subject of many studies, see, e.g., [9][10][11][12], because they have a variety of group-like properties. Notably, in [13], the authors define the commutativity degree of complete hypergroups and characterize it with an identity that is analogous to the class equation for groups. Recall that the commutativity degree of a finite group G was defined by W. Gustafson in [14] as the probability that two randomly chosen elements commute,…”

“…Inspired by this concept, in [13] the commutativity degree of a finite hypergroup (H, •) is defined as…”

The Reducibility Concept in General Hyperrings

“…This is why we will consider only proper complete hypergroups, i.e., complete hypergroups that are not groups. The heart ω H of a complete hypergroup (H, •) has an interesting property: it coincides with the set of identities of H. The complete hypergroups have been studied for their general properties [19], or in connection with their fuzzy grade [20], for their commutativity degree [21], or in relation with their size [22].…”