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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-1-2,
author = {Tetsuya Ishiu and Justin Tatch Moore},
title = {Minimality of non-$\sigma$-scattered orders},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {29-44},
zbl = {1209.03036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-1-2}
}
Tetsuya Ishiu; Justin Tatch Moore. Minimality of non-σ-scattered orders. Fundamenta Mathematicae, Tome 205 (2009) pp. 29-44. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-1-2/