A Simple Proof of Suzumura's Extension Theorem for Finite Domains With Applications

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.