Geodesic
On maximal actions and $w$-maximal actions of finite hypergroups
Journal of Algebraic Combinatorics, Tome 27 (2008) no. 2, pp. 127-141.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $\pi : K\rightarrow $Aff $( X)$, we will prove that $\pi $ is faithful and $| X|\geq | K|$, 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.
Keywords: keywords hypergroups, association schemes, actions, maximal actions, *-actions, $w$-maximal actions 
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     author = {Xu, Bangteng},
     title = {On maximal actions and $w$-maximal actions of finite hypergroups},
     journal = {Journal of Algebraic Combinatorics},
     pages = {127--141},
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Xu, Bangteng. On maximal actions and $w$-maximal actions of finite hypergroups. Journal of Algebraic Combinatorics, Tome 27 (2008) no. 2, pp. 127-141. http://geodesic.mathdoc.fr/item/JAC_2008__27_2_a7/