Interval Orders and Reverse Mathematics

We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2 ⊕ 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it contains neither 2 ⊕ 2 nor 3 ⊕ 1. Interval orders are a particular kind of partial orders which occur quite naturally in many different areas and have been widely studied. A partial order P = (P, ≤ P) is an interval order if the elements of P can be mapped to nonempty intervals of a linear order L so that p < P q holds iff every element of the interval associated to p precedes every element of the interval associated to q. The linear order L and the map from P to intervals are called an interval representation of P. The basic reference on interval orders is Fishburn's monograph [9]. The name " interval order " was introduced by Fishburn ([8]), although the notion was already studied much earlier by Norbert Wiener ([23]), who used the terminology " relation of complete sequence ". Interval orders model many phenomena occurring in the applied sciences: [9, §2.1] include examples such as chronological dating in archaeology and paleontology, scheduling of manufacturing processes, and psychophysical perception of sounds. Notice that if P is a countable interval order then we can assume that L is the rational or (as usual in applications) the real line (a real representation, in the terminology of [9]). Most recent research on interval orders (see e.g. the survey [22] and chapter 8 of [18]) focuses on finite partial orders, while in this paper we consider mostly infinite ones (although a careful analysis of the finite case is instrumental in obtaining results in the infinite case). A recent result about infinite interval orders shows that every interval order which is a well quasi-order is a better quasi-order ([15]). The basic characterization for interval orders is given by the following theorem proved independently by Fishburn ([8]) and Mirkin ([13]): Characterization Theorem 1. A partial order is an interval order if and only if it does not contain 2 ⊕ …