2015
DOI: 10.1007/s11856-015-1199-z
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Symmetrically complete ordered sets abelian groups and fields

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 symmet… Show more

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Cited by 7 publications
(20 citation statements)
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“…In [36] it is shown that a symmetrically complete ordered abelian group is spherically complete w.r.t. its natural valuation and hence a Hahn product, with all of its archimedean components isomorphic to R. Similarly, a symmetrically complete ordered field is spherically complete w.r.t.…”
Section: Cut Cofinalitiesmentioning
confidence: 99%
“…In [36] it is shown that a symmetrically complete ordered abelian group is spherically complete w.r.t. its natural valuation and hence a Hahn product, with all of its archimedean components isomorphic to R. Similarly, a symmetrically complete ordered field is spherically complete w.r.t.…”
Section: Cut Cofinalitiesmentioning
confidence: 99%
“…We say that (G, <) is symmetrically complete if (G, B) is spherically complete. In [6] we have characterized symmetrically complete ordered abelian groups and fields. We showed that every ordered abelian group (or field) can be extended to a symmetrically complete ordered abelian group (or field, respectively), and that all symmetrically complete ordered abelian groups are Hahn products with its archimedean components equal to R; it follows that they are divisible and therefore Q-vector spaces.…”
Section: An Application To Ordered Abelian Groups and Fieldsmentioning
confidence: 99%
“…The first and the second author have developed a simple approach that allows to extract common principles of proof for fixed point theorems in ultrametric spaces, metric spaces, ordered abelian groups and fields, topology, partially ordered sets, and lattices (see [3], [4], [5], and [6]). In this paper we will do the same for coincidence points and illustrate applications to the first three of the above mentioned areas.…”
Section: Introductionmentioning
confidence: 99%
“…But fortunately, this is not the case. In [12], Saharon Shelah has shown that every ordered field is contained in one that has a spherically complete order ball space (see also [7] for a power series field construction of such fields). So there are arbitrarily large ordered fields (and hence also ordered abelian groups) in which our above fixed point theorem holds.…”
Section: Proofmentioning
confidence: 99%