A family of totally ordered groups with some special properties
Annales mathématiques Blaise Pascal, Volume 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
Elena Olivos 1

1 Universidad de la Frontera Departamento de Matemática y Estadística Casilla 54-D Temuco Chile
@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/}
}
TY  - JOUR
AU  - Elena Olivos
TI  - A family of totally ordered groups with some special properties
JO  - Annales mathématiques Blaise Pascal
PY  - 2005
DA  - 2005///
SP  - 79
EP  - 90
VL  - 12
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.196/
UR  - https://www.ams.org/mathscinet-getitem?mr=2126442
UR  - https://zbmath.org/?q=an%3A1085.06010
UR  - https://doi.org/10.5802/ambp.196
DO  - 10.5802/ambp.196
LA  - en
ID  - AMBP_2005__12_1_79_0
ER  - 
%0 Journal Article
%A Elena Olivos
%T A family of totally ordered groups with some special properties
%J Annales mathématiques Blaise Pascal
%D 2005
%P 79-90
%V 12
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.196
%R 10.5802/ambp.196
%G en
%F AMBP_2005__12_1_79_0
Elena Olivos. A family of totally ordered groups with some special properties. Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 1, pp. 79-90. doi : 10.5802/ambp.196. https://ambp.centre-mersenne.org/articles/10.5802/ambp.196/

[1] W. Schikhof H. Ochsenius 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] 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

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

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

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

Cited by Sources: