Regular Actions of (Twin) Coxeter Hypergroups

Paul-Hermann Zieschang
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Paul-Hermann Zieschang: University of Texas Rio Grande Valley, School of Mathematical & Statistical Sciences

Chapter 10 in Hypergroups, 2023, pp 319-384 from Springer

Abstract: Abstract In this chapter we will show that both, the notion of a building in the sense of [44] or [45] and (modulo a certain building theoretic hypothesis) the notion of a twin building in the sense of [46], can be considered in a natural way as components of a theory of hypergroups. In Sections 10.2 and 10.3, we will see that semiregular buildings (as they will be defined in Section 10.1) and regular actions of Coxeter hypergroups are equivalent mathematical objects. The precise nature of this equivalence will be described in Theorems 10.2.1, 10.2.7, 10.3.1, 10.3.2, 10.3.3, and 10.3.4. In Sections 10.6 and 10.7, we will see that (modulo the above-mentioned hypothesis) thick twin buildings and regular actions of twin Coxeter hypergroups over sets of nonthin involutions are equivalent mathematical objects. The precise nature of this equivalence will be described in Theorems 10.6.1, 10.6.7, 10.7.1, 10.7.2, 10.7.3, and 10.7.4.

Date: 2023
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DOI: 10.1007/978-3-031-39489-8_10

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