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

Abstract

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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 ℵ κ , there exists a metrizable field K whose value group has cardinality ℵ κ . The existence of a cofinal sequence only depends on the choice of some appropriate ordinal α which has cardinality ℵ 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 ℵ κ there exists a metrizable field K of cardinality ℵ κ which has an absolutely convex subset that is not countably generated as a B K -module.We prove also that for any cardinality ℵ κ > ℵ 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.

How to cite

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Olivos, 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 $\aleph _\kappa &gt;\aleph _0$ 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 $\aleph _\kappa &gt;\aleph _0$ 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

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  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), 233-293 Zbl0938.46056MR1703500
  2. 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
  3. T. Jech, Set Theory, (1978), San Diego Academic Press, USA Zbl0419.03028MR506523
  4. P. Ribenboim, Théorie des valuations, (1968), Les Presses de l’Université de Montréal, Montréal, Canada Zbl0139.26201
  5. P. Ribenboim, The theory of classical valuations, (1998), Springer-Verlag Zbl0957.12005MR1677964

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