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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
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