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 || and value group G{1}, 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 κ , there exists a metrizable field K whose value group has cardinality κ . The existence of a cofinal sequence only depends on the choice of some appropriate ordinal α which has cardinality κ and which has cofinality ω.

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 κ there exists a metrizable field K of cardinality κ which has an absolutely convex subset that is not countably generated as a B K -module.

We prove also that for any cardinality κ > 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.

DOI : 10.5802/ambp.196
Olivos, Elena 1

1 Universidad de la Frontera Departamento de Matemática y Estadística Casilla 54-D Temuco Chile
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Olivos, Elena. 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. http://www.numdam.org/articles/10.5802/ambp.196/

[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 (1999), pp. 233-293 | MR | Zbl

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

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

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

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

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