Minimality of non-$\sigma$-scattered orders

Tetsuya Ishiu¹ ; Justin Tatch Moore²

¹Department of Mathematics and Statistics Miami University Oxford, OH 45056, U.S.A.
²Department of Mathematics Cornell University 310 Malott Hall Ithaca, NY 14853-4201, U.S.A.

Fundamenta Mathematicae, Tome 205 (2009) no. 1, pp. 29-44

Résumé

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.

Détail

DOI : 10.4064/fm205-1-2

Keywords: characterize under appropriate axiomatic assumptions linear order minimal respect being countable union scattered suborders assuming pfa only linear orders which minimal respect being sigma scattered either countryman types real types outline plausible approach demonstrating relative consistency there minimal non sigma scattered linear orders process establishing these results prove combinatorial characterizations given linear order sigma scattered contains either real aronszajn type