Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

Every binary relation ρρ on a set H,(card(H)>1) can define a hypercomposition and thus endow HH with a hypercompositional structure. In this paper, binary relations are represented by Boolean matrices. With their help, the hypercompositional structures (hypergroupoids, hypergroups, join hypergroups) that emerge with the use of the Rosenberg’s hyperoperation are characterized, constructed and enumerated using symbolic manipulation packages. Moreover, the hyperoperation given by x∘x={z∈H|(z,x)∈ρ} and x∘y=x∘x∪y∘yx∘y=x∘x∪y∘y is introduced and connected to Rosenberg’s hyperoperation, which assigns to every (x,y)(x,y) the set of all zz such that either (x,z)∈ρ(x,z)∈ρ or (y,z)∈ρ(y,z)∈ρ.