A family of totally ordered groups with some special properties
Annales Mathématiques Blaise Pascal, Tome 12 (2005) no. 1, pp. 79-90.

Let $K$ be a field with a Krull valuation $|\phantom{\rule{0.277778em}{0ex}}|$ and value group $G\ne \left\{1\right\}$, and let ${B}_{K}$ be the valuation ring. Theories about spaces of countable type and Hilbert-like spaces in [1] and spaces of continuous linear operators in [2] require that all absolutely convex subsets of the base field $K$ should be countably generated as ${B}_{K}$-modules.

By [1] Prop. 1.4.1, the field $K$ is metrizable if and only if the value group $G$ has a cofinal sequence. We prove that for any fixed cardinality ${\aleph }_{\kappa }$, there exists a metrizable field $K$ whose value group has cardinality ${\aleph }_{\kappa }$. The existence of a cofinal sequence only depends on the choice of some appropriate ordinal $\alpha$ which has cardinality ${\aleph }_{\kappa }$ and which has cofinality $\omega$.

By [2] Prop. 1.4.4, the condition that any absolutely convex subset of $K$ be countably generated as a ${B}_{K}$-module is equivalent to the fact that the value group has a cofinal sequence and each element in the completion ${G}^{#}$ is obtained as the supremum of a sequence of elements of $G$. We prove that for any fixed uncountable cardinal ${\aleph }_{\kappa }$ there exists a metrizable field $K$ of cardinality ${\aleph }_{\kappa }$ which has an absolutely convex subset that is not countably generated as a ${B}_{K}$-module.

We prove also that for any cardinality ${\aleph }_{\kappa }>{\aleph }_{0}$ 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.

@article{AMBP_2005__12_1_79_0,
author = {Elena Olivos},
title = {A family of totally ordered groups with some special properties},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {79--90},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {12},
number = {1},
year = {2005},
doi = {10.5802/ambp.196},
mrnumber = {2126442},
zbl = {1085.06010},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.196/}
}
Elena Olivos. A family of totally ordered groups with some special properties. Annales Mathématiques Blaise Pascal, Tome 12 (2005) no. 1, pp. 79-90. doi : 10.5802/ambp.196. https://ambp.centre-mersenne.org/articles/10.5802/ambp.196/

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[2] W. Schikhof H. Ochsenius Lipschitz operators in Banach spaces over Krull valued fields, Report N. 0310, University of Nijmegen, The Netherlands, Volume 13 (2003) | Zbl 1094.46049

[3] T. Jech Set Theory, San Diego Academic Press, USA, 1978 | MR 506523 | Zbl 0419.03028

[4] P. Ribenboim Théorie des valuations, Les Presses de l’Université de Montréal, Montréal, Canada, 1968 | Zbl 0139.26201

[5] P. Ribenboim The theory of classical valuations, Springer-Verlag, 1998 | MR 1677964 | Zbl 0957.12005

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