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

Athanasios Andrikopoulos

doi:10.1155/2019/5397036

Abstract

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.

Highlights

One of the most fundamental results on extensions of binary relations is due to Szpilrajn [1], who shows that any transitive and asymmetric binary relation has a transitive, asymmetric, and complete extension
One way of assessing whether a preference relation is rational is to check whether it can be extended to a transitive and complete relation. Another example is the problem of the existence of maximal elements of binary relations
In a very general sense, if ≽ is defined in a topological space, the existence of a linear extension of ≽ satisfying some continuity conditions is equivalent to the existence of a continuous utility function representing ≽ (a binary relation ≽ defined on X is represented by a utility function u : X 󳨀→ R, if for all x, y ∈ X : x ≽ y ⇐⇒ u(x) ≥ u(y))

Summary

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

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. 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 give a characterization of the existence of a realizer of a binary relation defined on an abelian group. These results generalize the well-known extension theorems of Szpilrajn, Dushnik-Miller, and Fuchs

Introduction

Notations and Definitions

The Main Results