Journal of Applied Mathematics and Decision Sciences
Volume 6 (2002), Issue 3, Pages 183-190
doi:10.1155/S1173912602000111
A simple proof of Suzumura's extension theorem for finite domains
with applications
Indian Institute of Management, Ahmedabad, India and School of Economic and Business Sciences, University of Witwatersrand at Johannesburg, South Africa
Copyright © 2002 Somdeb Lahiri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.