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)].


20N20 Hypergroups
05C99 Graph theory