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Basic Facts

In: Hypergroups

Author

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  • Paul-Hermann Zieschang

    (University of Texas Rio Grande Valley, School of Mathematical & Statistical Sciences)

Abstract

This chapter contains basic results on hypergroups as they were defined at the beginning of the Preface. We will refer to most of these results throughout the text. In Section 1.1, we show that each hypergroup has only one neutral element and only one inversion function. In Section 1.2, we compile results on products of hypergroup elements, and in Section 1.3, we provide computational rules for products of subsets of hypergroups. In Section 1.4, we consider thin hypergroup elements, and in Section 1.5, we establish the group correspondence which was mentioned in the Preface. In Section 1.6, we introduce actions of hypergroups and present a few general results on regular actions, and in Section 1.7, we present conditions a hypergroup needs to satisfy in order to admit a regular action. In Section 1.8, we connect regular actions of hypergroups with association schemes.

Suggested Citation

  • Paul-Hermann Zieschang, 2023. "Basic Facts," Springer Books, in: Hypergroups, chapter 1, pages 1-27, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-39489-8_1
    DOI: 10.1007/978-3-031-39489-8_1
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