Symmetrically complete ordered sets, abelian groups and fields
Logic
2013-08-06 v1
Abstract
We characterize and construct linearly ordered sets, abelian groups and fields that are {\emph symmetrically complete}, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and fields are important because generalizations of Banach's Fixed Point Theorem hold in them. We prove that symmetrically complete ordered abelian groups and fields are divisible Hahn products and real closed power series fields, respectively. We show how to extend any given ordered set, abelian group or field to one that is symmetrically complete. A main part of the paper establishes a detailed study of the cofinalities in cuts.
Keywords
Cite
@article{arxiv.1308.0780,
title = {Symmetrically complete ordered sets, abelian groups and fields},
author = {Katarzyna and Franz-Viktor Kuhlmann and Saharon Shelah},
journal= {arXiv preprint arXiv:1308.0780},
year = {2013}
}