Minimality of non-$\sigma$-scattered orders
Volume 205 / 2009
Fundamenta Mathematicae 205 (2009), 29-44
MSC: 03E05, 03E75, 06A05.
DOI: 10.4064/fm205-1-2
Abstract
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 ${\rm PFA}^+$, the only linear orders which are minimal with respect to not being $\sigma$-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-$\sigma$-scattered linear orders. In the process of establishing these results, we will prove combinatorial characterizations of when a given linear order is $\sigma$-scattered and when it contains either a real or Aronszajn type.
