On residually thin hypergroups

The notion of a hypergroup (in the sense of [11]) 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. [12], [13], and [15]. In the present article, we investigate residually thin hypergroups, that is hypergroups H which contain closed subsets F0, …, Fn such that F0={1}, Fn=H, and, for each element i in {1,…,n}, Fi−1⊆Fi and Fi//Fi−1 is thin. In our first main result, we analyze the normal structure of residually thin hypergroups. The second main result of this note says that finite hypergroups are residually thin if all of their elements h satisfy hh⁎h={h}. Our investigation, in particular our focus on this latter condition, was inspired by a study of metathin association schemes; cf. [6], [7]. We therefore would like to dedicate this article to Mitsugu Hirasaka who initiated this study.