# A family of totally ordered groups with some special properties

Elena Olivos^{[1]}

- [1] Universidad de la Frontera Departamento de Matemática y Estadística Casilla 54-D Temuco Chile

Annales mathématiques Blaise Pascal (2005)

- Volume: 12, Issue: 1, page 79-90
- ISSN: 1259-1734

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topOlivos, Elena. "A family of totally ordered groups with some special properties." Annales mathématiques Blaise Pascal 12.1 (2005): 79-90. <http://eudml.org/doc/10515>.

@article{Olivos2005,

abstract = {Let $K$ be a field with a Krull valuation $|\; |$ and value group $G\ne \lbrace 1\rbrace $, 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.},

affiliation = {Universidad de la Frontera Departamento de Matemática y Estadística Casilla 54-D Temuco Chile},

author = {Olivos, Elena},

journal = {Annales mathématiques Blaise Pascal},

keywords = {metrizable field; Krull valuation; value group},

language = {eng},

month = {1},

number = {1},

pages = {79-90},

publisher = {Annales mathématiques Blaise Pascal},

title = {A family of totally ordered groups with some special properties},

url = {http://eudml.org/doc/10515},

volume = {12},

year = {2005},

}

TY - JOUR

AU - Olivos, Elena

TI - A family of totally ordered groups with some special properties

JO - Annales mathématiques Blaise Pascal

DA - 2005/1//

PB - Annales mathématiques Blaise Pascal

VL - 12

IS - 1

SP - 79

EP - 90

AB - Let $K$ be a field with a Krull valuation $|\; |$ and value group $G\ne \lbrace 1\rbrace $, 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.

LA - eng

KW - metrizable field; Krull valuation; value group

UR - http://eudml.org/doc/10515

ER -

## References

top- 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), 233-293 Zbl0938.46056MR1703500
- W. Schikhof H. Ochsenius, Lipschitz operators in Banach spaces over Krull valued fields, Report N. 0310, University of Nijmegen, The Netherlands 13 (2003) Zbl1094.46049
- T. Jech, Set Theory, (1978), San Diego Academic Press, USA Zbl0419.03028MR506523
- P. Ribenboim, Théorie des valuations, (1968), Les Presses de l’Université de Montréal, Montréal, Canada Zbl0139.26201
- P. Ribenboim, The theory of classical valuations, (1998), Springer-Verlag Zbl0957.12005MR1677964

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