Generalizations of Szpilrajn's Theorem in economic and game theories

Szpilrajn's Lemma entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn's Lemma to show that each partial order has a relizer. Since then, many authors utilize Szpilrajn's Theorem and the Well-ordering principle to prove more general existence type theorems on extending binary relations. Nevertheless, we are often interested not only in the existence of extensions of a binary relation $R$ satisfying certain axioms of orderability, but in something more: (A) The conditions of the sets of alternatives and the properties which $R$ satisfies to be inherited when one passes to any member of a subfamily of the family of extensions of $R$ and: (B) The size of a family of ordering extensions of $R$, whose intersection is $R$, to be the smallest one. The key to addressing these kinds of problems is the szpilrajn inherited method. In this paper, we define the notion of $\Lambda(m)$-consistency, where $m$ can reach the first infinite ordinal $\omega$, and we give two general inherited type theorems on extending binary relations, a Szpilrajn type and a Dushnik-Miller type theorem, which generalize all the well known existence and inherited type extension theorems in the literature. \keywords{Consistent binary relations, Extension theorems, Intersection of binary relations.