Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

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