A simple proof of Suzumura's extension theorem for finite domains with applications

Somdeb Lahiri

doi:10.1155/s1173912602000111

Abstract

In this paper we provide a simple proof of the extension theorem for partial orderings due to Suzumura [1983] when the domain of the partial order is finite. The extension theorem due to Szpilrajn [1930] follows from this theorem. Szpilrajns extension theorem is used to show that an asymmetric binary relation is contained in the asymmetric part of a linear order if and only if it is acyclic. This theorem is then applied to prove three results. Finally we introduce the concept of a threshold choice function, and our third result says that such choice functions are the only ones to satisfy a property called functional acyclicity.

Highlights

In this paper we provide a simple proof of the extension theorem for partial orderings due to Suzumura [1983] when the domain of the partial order is finite
The well known result due to Dushnik and Miller [1941], which states that any asymmetric and transitive binary relation is the intersection of linear orders follows as an immediate corollary of this result
The second result is a theorem in Lahiri [1999], which says that a choice function is a batch choice function if and only if it satisfies a property called the choice acyclicity property

Summary

Introduction

In this paper we provide a simple proof of the extension theorem for partial orderings due to Suzumura [1983] when the domain of the partial order is finite. Szpilrajn’s extension theorem is used to show that an asymmetric binary relation is contained in the asymmetric part of a linear order if and only if it is acyclic. This theorem is applied to prove three results. We introduce the concept of a threshold choice function, and our third result says that such choice functions are the only ones to satisfy a property called functional acyclicity. This last property can be traced to Aizerman and Aleskerov [1995] as well

The Extension Theorems

Batch Choice Functions

Functional Acyclicity