About the fundamental relations defined on the hypergroupoids associated with binary relations. (Q607368)

The paper deals with connections between hypergroupoids and \(n\)-ary relations. The authors show that the hypergroupoid associated with an \(n\)-ary relation [as in \textit{I. G. Rosenberg}, Ital. J. Pure Appl. Math. 4, 93-101 (1998; Zbl 0962.20055), or \textit{P. Corsini}, Ital. J. Pure Appl. Math. 7, 11-18 (2000; Zbl 0972.20042), or \textit{I. Cristea, M. Ştefănescu}, Eur. J. Comb. 31, No. 3, 780-789 (2010; Zbl 1194.20068)], coincides with the one associated with a certain induced binary relation. On the basis of the properties of some fundamental relations defined on a hypergroupoid, they determine necessary and sufficient conditions for two elements in a hypergroupoid associated with a binary relation to be operationally equivalent or inseparable. They present some links with the Boolean matrices associated with binary relations and determine conditions for a Boolean matrix \(M(\rho)\) in order to have that the hypergroupoid associated with \(\rho\) is a reduced hypergroup. Also, they characterize the matrix representing a binary relation such that its associated hypergroupoid is an \(H_v\)-group.