Regular actions of Coxeter hypergroups

AbstractGeneralizing the notion of a group action, we define actions of hypergroups. Our focus is on regular actions. In contrast to groups which always allow exactly one (right) regular action hypergroups may have no regular action, they also may have infinitely many regular actions. With an eye on the second case we establish a class of hypergroups the regular actions of which correspond in a bijective way to semiregular buildings. (The class of semiregular buildings includes the class of thick buildings and is defined via a regularity condition on the panels of a building.)KEYWORDS: Coxeter groupsCoxeter hypergroupsgroup theoryhypergroupsregular actions2020 MATHEMATICS SUBJECT CLASSIFICATION: 51E2420N2043A62 AcknowledgmentsThe authors would like to thank an anonymous referee for useful comments on the first submission of this article.Notes1 We work here with a slight modification of Marty’s definition, we require the existence of a neutral element. Our notion of a hypergroup is then the same as the one given in [3, 4], and [8], and is equivalent to the notion of a polygroup in the sense of Bijan Davvaz; cf. [2, Definition 3.1.1].2 Semiregular buildings are defined in Section 9. The definition implies that thick buildings are semiregular.3 We notice that, for each hypergroup H and any two elements a and b of H, the product ab is not empty. This is, in fact easy to see. From [8, Lemma 2.1] we know that 1∈b**b*, from [3, Lemma 2.1(ii)] that b**=b*. Thus, 1∈bb*. On the other hand, we have a∈a1. It follows that a∈abb*, and that shows that ab is not empty. We will use this observation occasionally without special mention.4 It is easy to see that, via the group correspondence, thin Coxeter hypergroups correspond to Coxeter groups.5 Our definition is similar to the definition of an action of a polygroup as suggested in [2, Definition 3.7.1].6 Our definition coincides with the one given in [2, Definition 2.4.1(3)].7 By a Coxeter system we mean a pair consisting of a group W and a set of involutions, so that, via the group correspondence, W is a thin Coxeter hypergroup over I. This definition is equivalent to the one given in [6].8 While the definition of a thick building is common in the literature, the one of a semiregular building is not.9 The ω-canonical hypergroup isomorphism was introduced right after Theorem 3.8.