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1996 | 151 | 2 | 177-187
Tytuł artykułu

Monotone σ-complete groups with unbounded refinement

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
151
Numer
2
Strony
177-187
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-12-19
Twórcy
Bibliografia
  • [1] A. Bigard, K. Keimel et S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math. 608, Springer, 1977.
  • [2] R. Bradford, Cardinal addition and the axiom of choice, Ann. Math. Logic 3 (1971), 111-196.
  • [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.
  • [6] P. A. Fillmore, The dimension theory of certain cardinal algebras, Trans. Amer. Math. Soc. 117 (1965), 21-36.
  • [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 $ℵ_0$-continuous regular rings, Mem. Amer. Math. Soc. 234 (1980).
  • [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.
  • [11] F. Wehrung, Metric properties of positively ordered monoids, Forum Math. 5 (1993), 183-201.
  • [12] F. Wehrung, Non-measurability properties of interpolation vector spaces, preprint.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv151i2p177bwm
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