Abstract
We characterize linearly ordered sets, abelian groups and fields that are 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. This gives us a direct route to the construction of symmetrically complete ordered abelian groups and fields, modulo an analogous construction at the level of ordered sets; in particular, this gives an alternative approach to the construction of symmetrically complete fields in [12].
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The research of the second author was partially supported by a Canadian NSERC grant and a sabbatical grant from the University of Saskatchewan.
The third author would like to thank the Israel Science Foundation for partial support of this research (Grant no. 1053/11). Paper 1024 on his publication list.
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Kuhlmann, K., Kuhlmann, FV. & Shelah, S. Symmetrically complete ordered sets abelian groups and fields. Isr. J. Math. 208, 261–290 (2015). https://doi.org/10.1007/s11856-015-1199-z
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DOI: https://doi.org/10.1007/s11856-015-1199-z