Monotone σ-complete groups with unbounded refinement

Friedrich Wehrung

Fundamenta Mathematicae (1996)

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

Abstract

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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 ∑ m a m = ∑ n b 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.

How to cite

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Wehrung, 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

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

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