# Monotone σ-complete groups with unbounded refinement

Fundamenta Mathematicae (1996)

- Volume: 151, Issue: 2, page 177-187
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topWehrung, Friedrich. "Monotone σ-complete groups with unbounded refinement." Fundamenta Mathematicae 151.2 (1996): 177-187. <http://eudml.org/doc/212189>.

@article{Wehrung1996,

abstract = {The real line ℝ may be characterized as the unique non-atomic directed partially ordered abelian group which is monotone σ-complete (countable increasing bounded sequences have suprema), has the countable refinement property (countable sums $∑_ma_m = ∑_nb_n$ of positive (possibly infinite) elements have common refinements) and is linearly ordered. We prove here that the latter condition is not redundant, thus solving an old problem by A. Tarski, by proving that there are many spaces (in particular, of arbitrarily large cardinality) satisfying all the above listed axioms except linear ordering.},

author = {Wehrung, Friedrich},

journal = {Fundamenta Mathematicae},

keywords = {monotone σ-complete groups; partially ordered vector spaces; Archimedean condition; countable refinement property; directed Archimedean partially ordered Abelian group; monotone -complete cofinal embedding; cardinal space; nonlinearly ordered cardinal groups},

language = {eng},

number = {2},

pages = {177-187},

title = {Monotone σ-complete groups with unbounded refinement},

url = {http://eudml.org/doc/212189},

volume = {151},

year = {1996},

}

TY - JOUR

AU - Wehrung, Friedrich

TI - Monotone σ-complete groups with unbounded refinement

JO - Fundamenta Mathematicae

PY - 1996

VL - 151

IS - 2

SP - 177

EP - 187

AB - The real line ℝ may be characterized as the unique non-atomic directed partially ordered abelian group which is monotone σ-complete (countable increasing bounded sequences have suprema), has the countable refinement property (countable sums $∑_ma_m = ∑_nb_n$ of positive (possibly infinite) elements have common refinements) and is linearly ordered. We prove here that the latter condition is not redundant, thus solving an old problem by A. Tarski, by proving that there are many spaces (in particular, of arbitrarily large cardinality) satisfying all the above listed axioms except linear ordering.

LA - eng

KW - monotone σ-complete groups; partially ordered vector spaces; Archimedean condition; countable refinement property; directed Archimedean partially ordered Abelian group; monotone -complete cofinal embedding; cardinal space; nonlinearly ordered cardinal groups

UR - http://eudml.org/doc/212189

ER -

## References

top- [1] A. Bigard, K. Keimel et S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math. 608, Springer, 1977. Zbl0384.06022
- [2] R. Bradford, Cardinal addition and the axiom of choice, Ann. Math. Logic 3 (1971), 111-196. Zbl0287.02041
- [3] R. Chuaqui, Simple cardinal algebras, Notas Mat. Univ. Católica de Chile 6 (1976), 106-131.
- [4] A. B. Clarke, A theorem on simple cardinal algebras, Michigan Math. J. 3 (1955-56), 113-116.
- [5] A. B. Clarke, On the representation of cardinal algebras by directed sums, Trans. Amer. Math. Soc. 91 (1959), 161-192. Zbl0085.25904
- [6] P. A. Fillmore, The dimension theory of certain cardinal algebras, Trans. Amer. Math. Soc. 117 (1965), 21-36. Zbl0146.01803
- [7] K. R. Goodearl, Partially Ordered Abelian Groups with Interpolation, Math. Surveys Monographs 20, Amer. Math. Soc., 1986.
- [8] K. R. Goodearl, D. E. Handelman and J. W. Lawrence, Affine representations of Grothendieck groups and applications to Rickart C*-algebras and ${\aleph}_{0}$-continuous regular rings, Mem. Amer. Math. Soc. 234 (1980). Zbl0435.16005
- [9] A. Tarski, Cardinal Algebras, Oxford Univ. Press, New York, 1949.
- [10] F. Wehrung, Injective positively ordered monoids I, J. Pure Appl. Algebra 83 (1992), 43-82. Zbl0790.06016
- [11] F. Wehrung, Metric properties of positively ordered monoids, Forum Math. 5 (1993), 183-201. Zbl0769.06008
- [12] F. Wehrung, Non-measurability properties of interpolation vector spaces, preprint. Zbl0916.06018

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.