An Extension Theorem for Non-Transitive Preferences

In his classical contribution Szpilrajn (1930) shows that any quasi-ordering (a reflexive and transitive binary relation) can be extended to an ordering (a complete quasi-ordering). The purpose of this paper is to provide an analogous extension result for non-transitive preferences. The case of quasi-transitivity (transitivity of the strict preference) is trivial since it suffices to `complete' the original preference by putting indifference for all pairs of alternatives that are non-comparable. The focus of our analysis is a different notion of non-transitivity called $sigma$-transitivity and put forward bySen (1970).Our main result is an extension theorem which shows that any $sigma$-transitive preference can be extended to a complete preference preserving$sigma$-transitivity. We also provide the dual result in terms of a strict preference approach.Finally, we show an application to the theory of choice by providing a new characterization of the Weak Axiom of Revealed Preference.