Generalizations of Szpilrajn's Theorem in economic and game theories

Abstract

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 (m)-consistency, where m can reach the first infinite ordinal , 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. Consistent binary relations, Extension theorems, Intersection of binary relations.

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Reproductions