Hypergroups and binary relations
A hypergroupoid is a set \(H\neq\emptyset\) endowed with a function \(\circ\) from \(H\times H\) to the set of non-empty subsets of \(H\). A hypergroup is a hypergroupoid \((H,\circ)\) such that \(x\circ(y\circ z)=(x\circ y)\circ z\) and \(x\circ H=H=H\circ x\) for all \(x,y,z\in H\). To a binary relation \(R\) on \(H\), a hypergroupoid \(H_R\) is associated as follows: \(x\circ x=\{y\in H;\;(x,y)\in R\}\) and \(x\circ y=x\circ x\cup y\circ y\). I. G. Rosenberg found conditions on \(R\) such that \(H_R\) is a hypergroup. The first author studied these hypergroups under the union, intersection, Cartesian product, product and direct limit of relations. In this paper this treatment is continued. The authors obtain several results for which mutual associativity plays a role and describe hypergroups associated with trees.
scientific article; zbMATH DE number 1899942