A Note on Numerical Representations of Quasi-Transitive Binary Relations

A Binary relation is said to be quasi-transitive if its asymmetric part is transitive. A binary relation is said to be a weak-order if it is reflexive, complete and transitive. One may refer to Frensh [1986] for a synopsis of these and similar definitions. An easy consequence of the result in Donaldson and Weymark [1998], which states that, any binary relation which is reflexive, complete and quasi-transitive can be expressed as an intersection of weak orders, is the result (which for a finite domain may be traced back to Aizerman and Malishevsky [1981], [see Aizerman and Aleskerov [1995] as well]) that the asymmetric part of a quasi-transitive binary relation can be expressed as the intersection of the asymmetric parts of weak-orders. In this note we provide a new and an independent proof of this result (which is what we refer to as Theorem I in this note) considering its abiding importance in decision theory. We also, use our Theorem I to prove the well known result due to Dushnik and Miller [1941], which states that any asymmetric and transitive binary relation is the intersection of binary relations which are asymmetric, transitive and complete. Finally, we provide a new proof of the result due to Donaldson and Weymark [1998], mentioned. Our proof appears to be simpler than the one provided by them, and is established with the help of the Theorem due to Dushnik and Miller [1941].