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A family of totally ordered groups with some special properties. (English) Zbl 1085.06010

In this paper the following main results are obtained. a) It is proved that for any fixed cardinality \({\aleph}_k\) there exists a metrizable field \(K\) whose value group has cardinality \({\aleph}_k\). b) It is proved that for any fixed uncountable cardinal \({\aleph}_k\) there exists a metrizable field \(K\) of cardinality \({\aleph}_k\) which has an absolutely convex subset that is not countably generated as a \(B_K\)-module. c) It is proved that for any cardinality \({\aleph}_k > {\aleph}_0\) for the value group the two conditions (the whole group has a cofinal sequence and every subset of the group which is bounded above has a cofinal sequence) are logically independent.

MSC:

06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
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References:

[1] H. Ochsenius, W. Schikhof, Banach spaces over fields with an infinite rank valuation, In p-Adic Functional Analysis, Lecture Notes in pure and applied mathematics 207, edited by J. Kakol, N. De Grande-De Kimpe and C. Pérez García. Marcel Dekker, 233-293 (1999) · Zbl 0938.46056
[2] H. Ochsenius, W. Schikhof, Lipschitz operators in Banach spaces over Krull valued fields, Report N. 0310, University of Nijmegen, The Netherlands, 13 (2003) · Zbl 1094.46049
[3] Jech, T., Set Theory (1978) · Zbl 0419.03028
[4] Ribenboim, P., Théorie des valuations (1968) · Zbl 0139.26201
[5] Ribenboim, P., The theory of classical valuations (1998) · Zbl 0957.12005
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