Minimality of non-σ-scattered orders

Abstract

In this paper we will characterize — under appropriate axiomatic assumptions — when a linear order is minimal with respect to not being a countable union of scattered suborders. We show that, assuming PFA, the only linear orders which are minimal with respect to not being σ-scattered are either Countryman types or real types. We also outline a plausible approach to demonstrating the relative consistency of There are no minimal non σ-scattered linear orders. In the process of establishing these results, we will prove combinatorial characterizations of when a given linear order is σ-scattered and when it contains either a real or Aronszajn type.