### Symmetrically complete ordered sets, abelian groups and fields

by Kuhlmann and Kuhlmann and Shelah. [KuKuSh:1024]

Israel J Math, 2015

We characterize and construct linealy ordered sets, abelian groups
and fields that are ph 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 can be proved
for 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.

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