Rings arising from finite tight hypergroups

Abstract

The notion of a hypergroup (in the sense of [8]) provides a far reaching and meaningful generalization of the concept of a group. Specific classes of hypergroups have given rise to challenging questions and interesting connections to geometric and group theoretic topics; cf. [4], [9], [10], and [14]. The present article is a continuation of a study of residually thin hypergroups which was initiated in [4]. One of the main results of [4] says that finite tight hypergroups, that is hypergroups all elements h of which satisfy hh⁎h={h}, are residually thin; cf. [4, Theorem 7.7]. (Tight hypergroups occur in the study of metathin association schemes; cf. [3,5–7,12,13].) In the present article, we take advantage of this result in order to construct, for each finite tight hypergroup satisfying a mild extra condition, an associative ring. The rings which we find generalize the construction of scheme rings for a certain class of association schemes, including those which correspond to finite groups (group rings).