On maximal actions and <Emphasis Type="Italic">w</Emphasis>-maximal actions of finite hypergroups |
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Authors: | Bangteng Xu |
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Institution: | (1) Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475, USA |
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Abstract: | Sunder and Wildberger (J. Algebr. Comb.
18, 135–151, 2003) introduced the notion of actions of finite hypergroups, and studied maximal irreducible actions and *-actions. One of the
main results of Sunder and Wildberger states that if a finite hypergroup K admits an irreducible action which is both a maximal action and a *-action, then K arises from an association scheme. In this paper we will first show that an irreducible maximal action must be a *-action,
and hence improve Sunder and Wildberger’s result (Theorem 2.9). Another important type of actions is the so-called w-maximal actions. For a w-maximal action π:K→Aff (X), we will prove that π is faithful and |X|≥|K|, and |K| is the best possible lower bound of |X|. We will also discuss the strong connectivity of the digraphs induced by a w-maximal action. |
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Keywords: | Hypergroups Association schemes Actions Maximal actions *-actions w-maximal actions |
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