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Hypergroups and join spaces determined by relations. (English) Zbl 0962.20055
Let $$\rho$$ be a binary relation on a set $$H$$. For $$x\in H$$ we put $$\rho(x)=\{y\in H\mid (x,y)\in\rho\}$$ and suppose that the relation $$\rho$$ is with full domain (i.e. $$\rho(x)$$ is not empty for all $$x\in H$$). Define a hyperoperation on $$H$$ by $$x\circ y=\rho(x)\cup\rho(y)$$. The purpose of this paper is to study the hypergroupoid $$H_\rho=(H,\circ)$$. The relations $$\rho$$ such that $$H_\rho$$ is, respectively a semihypergroup, a hypergroup, a join space, are characterized. Note that the previous construction generalizes some others obtained by J. Nieminen [J. Geom. 33, No. 1/2, 99-103 (1988; Zbl 0655.05058)], P. Corsini [J. Comb. Inf. Syst. Sci. 16, No. 4, 313-318 (1991; Zbl 0783.20041)] and J. Chvalina [General algebra and ordered sets, Proc. Summer School 1994, Olomouc (Czech Republic), 19-30 (1994; Zbl 0827.20085)].

##### MSC:
 20N20 Hypergroups 05C99 Graph theory