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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