Symmetrically complete ordered sets, abelian groups and fields

TL;DR

This paper characterizes and constructs symmetrically complete ordered sets, abelian groups, and fields, which are crucial for fixed point theorems and are shown to be divisible Hahn products and real closed power series fields.

Contribution

It provides a complete characterization and construction methods for symmetrically complete structures, linking them to well-known algebraic and ordered systems.

Findings

Symmetrically complete ordered abelian groups are divisible Hahn products.

Symmetrically complete ordered fields are real closed power series fields.

Extensions to symmetrically complete structures are systematically constructed.

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.