A Cantor-Bernstein theorem for σ -complete MV-algebras

Anna de Simone; Daniele Mundici; Mirko Navara

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 2, page 437-447
  • ISSN: 0011-4642

Abstract

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The Cantor-Bernstein theorem was extended to σ -complete boolean algebras by Sikorski and Tarski. Chang’s MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Łukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to σ -complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras.

How to cite

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de Simone, Anna, Mundici, Daniele, and Navara, Mirko. "A Cantor-Bernstein theorem for $\sigma $-complete MV-algebras." Czechoslovak Mathematical Journal 53.2 (2003): 437-447. <http://eudml.org/doc/30789>.

@article{deSimone2003,
abstract = {The Cantor-Bernstein theorem was extended to $\sigma $-complete boolean algebras by Sikorski and Tarski. Chang’s MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Łukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to $\sigma $-complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras.},
author = {de Simone, Anna, Mundici, Daniele, Navara, Mirko},
journal = {Czechoslovak Mathematical Journal},
keywords = {Cantor-Bernstein theorem; MV-algebra; boolean element of an MV-algebra; partition of unity; direct product decomposition; $\sigma $-complete MV-algebra; Cantor-Bernstein theorem; MV-algebra; Boolean element; partition of unity; direct product decomposition; -complete MV-algebra},
language = {eng},
number = {2},
pages = {437-447},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A Cantor-Bernstein theorem for $\sigma $-complete MV-algebras},
url = {http://eudml.org/doc/30789},
volume = {53},
year = {2003},
}

TY - JOUR
AU - de Simone, Anna
AU - Mundici, Daniele
AU - Navara, Mirko
TI - A Cantor-Bernstein theorem for $\sigma $-complete MV-algebras
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 2
SP - 437
EP - 447
AB - The Cantor-Bernstein theorem was extended to $\sigma $-complete boolean algebras by Sikorski and Tarski. Chang’s MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Łukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to $\sigma $-complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras.
LA - eng
KW - Cantor-Bernstein theorem; MV-algebra; boolean element of an MV-algebra; partition of unity; direct product decomposition; $\sigma $-complete MV-algebra; Cantor-Bernstein theorem; MV-algebra; Boolean element; partition of unity; direct product decomposition; -complete MV-algebra
UR - http://eudml.org/doc/30789
ER -

References

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  1. An invitation to Chang’s MV-algebras, In: Advances in Algebra and Model Theory, M.  Droste, R. Göbel (eds.), Gordon and Breach Publishing Group, Reading, UK, 1997, pp. 171–197. (1997) MR1683528
  2. Algebraic Foundations of Many-valued Reasoning. Trends in Logic. Vol.  7, Kluwer Academic Publishers, Dordrecht, 1999. (1999) MR1786097
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  4. 10.1023/A:1022467218309, Czechoslovak Math. J. 49(124) (1999), 517–526. (1999) MR1708370DOI10.1023/A:1022467218309
  5. A solution to a problem of Sikorski, Fund. Math. 40 (1953), 39–41. (1953) MR0060809
  6. Basic Set Theory. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1979. (1979) MR0533962
  7. 10.1016/0022-1236(86)90015-7, J.  Funct. Anal. 65 (1986), 15–63. (1986) Zbl0597.46059MR0819173DOI10.1016/0022-1236(86)90015-7
  8. Boolean Algebras, Springer-Verlag. Ergebnisse Math. Grenzgeb., Berlin, 1960. (1960) Zbl0087.02503MR0126393
  9. A generalization of a theorem of Banach and Cantor-Bernstein, Colloq. Math. 1 (1948), 140–144 and 242. (1948) MR0027264
  10. Cardinal Algebras, Oxford University Press, New York, 1949. (1949) Zbl0041.34502MR0029954

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