Hypergroups - Association Schemes - Buildings

Paul Hermann Zieschang
Depatment of Mathematics, University of Texas, Brownsville, TX 78520



Content: Let $S$ be a set, and let $\mu$ be a map from $S\times S$ to the power set of $S$. For any two elements $p$ and $q$ of $S$, we write $pq$ instead of $\mu(p,q)$ and assume that $pq$ is not empty. For any two non-empty subsets $P$ and $Q$ of $S$, we define the {\em complex product}\index{complex product} $PQ$ to be the union of the sets $pq$ with $p\in P$ and $q\in Q$. If one of the two factors in a complex product consists of a single element, say $s$, we write $s$ instead of $\{s\}$ in that product. Following (and generalizing) Fr$\acute{{\rm e}}$d$\acute{{\rm e}}$ric Marty's terminology (1934) we call $S$ a {\em hypergroup}\index{hypergroup} (with respect to $\mu$) if the following three conditions hold. \quad 1. $\forall$ $p$, $q$, $r\in S$: $p(qr)=(pq)r$. \quad 2. $\exists$ $e\in S$ $\forall$ $s\in S$: $se=\{s\}$. \quad 3. $\forall$ $s\in S$ $\exists$ $s^{*}\in S$ $\forall$ $p$, $q$, $r\in S$: $p\in q^{*}r^{*}$ $\Rightarrow$ $q\in r^{*}p^{*}$ and $r\in p^{*}q^{*}$. Each association scheme satisfies the above three conditions with respect to its complex multiplication. Thus, hypergroups generalize association schemes. I will explain how association schemes may take advantage of the structure theory of hypergroups. Special attention will be given to the embedding of the theory of buildings (also the definition of a building will be given in my talk) into scheme theory.

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