Enumeration of hypercompositional structures defined by binary relations

Abstract

This paper deals with hyperoperations that derive from binary relations and it studies the hypercompositional structures that are created by them.  It is proved that if ρ is a binary relation on a non-void set H, then the hypercomposition xy = {z ∈ H : (x, z) ∈ ρ and (z, y) ∈ ρ} satisfies the associativity or the reproductivity only when it is total.  There also appear routines that calculate (with the use of small computing power) the number of non isomorphic hypergroupoids, when the cardinality of H is finite.