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On the heart of hypergroups. (English) Zbl 0810.20057

Let $H$ be a hypergroup and consider on $H$ the relations ${\left({\beta }_{n}\right)}_{n\in {ℕ}^{*}}$ defined by: $x{\beta }_{1}y$ if and only if $x=y$ and if $n\ge 2$ then $x{\beta }_{n}y$ if and only if there exists $\left({x}_{1},\cdots ,{x}_{n}\right)\in {H}^{n}$ such that $x$ and $y$ are in ${x}_{1}{x}_{2}\cdots {x}_{n}$. The relation $\beta ={\bigcup }_{n\in {ℕ}^{*}}{\beta }_{n}$ is an equivalence relation on $H$, it is strongly regular and $H/\beta$ is a group [see P. Corsini, Prolegomena of Hypergroup Theory. (1993; Zbl 0785.20032)]. If 1 is the identity of the group $H/\beta$ and $p:H\to H/\beta$ is the canonical projection then ${\omega }_{H}={p}^{-1}\left(1\right)$ is called the heart of $H$.

The purpose of this paper is to characterize the hypergroups $H$ for which the heart is a hyperproduct, i.e. there exist $n\in {ℕ}^{*}$ and $\left({x}_{1},\cdots ,{x}_{n}\right)\in {H}^{n}$ such that ${\omega }_{H}={x}_{1}\cdots {x}_{n}$.

##### MSC:
 20N20 Hypergroups (group theory)