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On the partial semi-hypergroups with empty diagonal. (English) Zbl 0962.20054
Let $$H$$ be a set with $$n$$ elements and $$\circ\colon H\times H\to{\mathcal P}(H)$$ a partial hyperoperation on $$H$$. The purpose of this paper is to study the hypergroupoids $$(H,\circ)$$ which satisfy the conditions $$(x\circ y)\circ z=x\circ(y\circ z)$$ and $$x\circ x=\emptyset$$, for all $$x,y,z\in H$$.
Denote $$\alpha=\text{Card}\{(x,y)\in H^2\mid x\circ y\neq\emptyset\}$$, $$\beta=\max\{\text{Card}(x\circ y)\mid(x,y)\in H^2\}$$. The author proves that $$\alpha\leq n^2-2n$$ and $$\beta\leq n-2$$. All the hypergroupoids $$(H,\circ)$$ such that $$\beta=n-2$$ are found. Partial results for the cases $$\beta=n-3$$ or $$n\leq 5$$ are given.
##### MSC:
 20N20 Hypergroups 08A55 Partial algebras
##### Keywords:
partial hyperoperations; hypergroupoids
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