Szpilrajn-type theorems in economics

The Szpilrajn "constructive type" theorem on extending binary relations, or its generalizations by Dushnik and Miller [10], is one of the best known theorems in social sciences and mathematical economics. Arrow [1], Fishburn [11], Suzumura [22], Donaldson and Weymark [8] and others utilize Szpilrajn's Theorem and the Well-ordering principle to obtain more general "existence type" theorems on extending binary relations. Nevertheless, we are generally interested not only in the existence of linear extensions of a binary relation R, but in something more: the conditions of the preference sets and the properties which $R$ satisfies to be "inherited" when one passes to any member of some \textquotedblleft interesting\textquotedblright family of linear extensions of R. Moreover, in extending a preference relation $R$, the problem will often be how to incorporate some additional preference data with a minimum of disruption of the existing structure or how to extend the relation so that some desirable new condition is fulfilled. The key to addressing these kinds of problems is the szpilrajn constructive method. In this paper, we give two general "constructive type" theorems on extending binary relations, a Szpilrajn type and a Dushnik-Miller type theorem, which generalize and give a "constructive type" version of all the well known extension theorems in the literature.