Binary relations and hypergroupoids (Q2706753)

The author associates a partial hyperoperation \(\langle\widetilde\circ_R\rangle\) to every binary relation \(R\) defined on a non-empty set \(H\) in the following way: \(x\widetilde\circ_R y=\{z\in H\mid xRz,\;zRy\}\).NEWLINENEWLINENEWLINEThe hyperstructure \(\langle H,\widetilde\circ_R\rangle\) is a partial hypergroupoid and the necessary and sufficient condition so that \(\langle H,\widetilde\circ_R\rangle\) be a hypergroupoid is \(R\circ R=H\times H\). It is proved that \(\langle K,\widetilde\circ_R\rangle\) is a subquasihypergroup of \(H\) iff the hyperoperation \(\langle\widetilde\circ_R\rangle\) is total.NEWLINENEWLINENEWLINEConditions so that \(\langle\widetilde\circ_R\rangle\) is defined everywhere are presented. Especially, in the finite case, (\(\text{card }H=n\), \(n>1\)), it is shown that the previous is true iff \((M_R)^2=T\), where \(M_R\) is the matrix associated with the relation \(R\) (\(M_R=(a_{ij})\), \(a_{ij}\in\{0,1\}\)) and \(T=(t_{ij})\) with \(\forall(i,j)\), \(t_{ij}=1\). The complete answer with all related tables is given in the cases where \(\text{card }H=2,3\).NEWLINENEWLINENEWLINEFinally, a generalization \(\left(\left\langle\begin{smallmatrix} s\\ \circ\\ n\end{smallmatrix}\right\rangle\right)\) of the hyperoperation \(\langle\widetilde\circ\rangle\left(=\left\langle\begin{smallmatrix} 1\\ \circ\\ 2\end{smallmatrix}\right\rangle\right)\) is introduced.