A Generalization of Arrow’s Lemma on Extending a Binary Relation

Abstract EN

By examining whether the individualistic assumptions used in social choice could be used in the aggregation of individual preferences, Arrow proved a key lemma that generalizes the famous Szpilrajn’s extension theorem and used it to demonstrate the impossibility theorem.

In this paper, I provide a characterization of Arrow’s result for the case in which the binary relations I extend are not necessarily transitive and are defined on abelian groups.

I also give a characterization of the existence of a realizer of a binary relation defined on an abelian group.

These results also generalize the well-known extension theorems of Szpilrajn, Dushnik-Miller, and Fuchs.