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On a theorem of Cantor-Bernstein type for algebras

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Abstract

Freytes proved a theorem of Cantor-Bernstein type for algbras; he applied certain sequences of central elements of bounded lattices. The aim of the present paper is to extend the mentioned result to the case when the lattices under consideration need not be bounded; instead of sequences of central elements we deal with sequences of internal direct factors of lattices.

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References

  1. R. Cignoli, I. D. D’Ottaviano, and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht, 2000.

    MATH  Google Scholar 

  2. A. De Simone, D. Mundici, and M. Navara: A Cantor-Bernstein theorem for σ-complete MV-algebras. Czechoslovak Math. J. 53 (2003), 437–447.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. De Simone, M. Navara, and P. Pták: On interval homogeneous orthomodular lattices. Commentat. Math. Univ. Carolinae 42 (2001), 23–30.

    MATH  Google Scholar 

  4. A. Dvurečenskij: Central elements and Cantor-Bernstein’s theorem for pseudo-effect algebras. J. Aust. Math. Soc. 74 (2003), 121–143.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Freytes: An algebraic version of the Cantor-Bernstein-Schröder theorem. Czechoslovak Math. J. 54 (2004), 609–621.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Georgescu, A. Iorgulescu: Pseudo MV-algebras: a noncommutative extension of MV-algebras. Proc. Fourth Int. Symp. Econ. Informatics, Bucharest. 1999, pp. 961–968.

  7. G. Georgescu, A. Iorgulescu: Pseudo MV-algebras. Multiple-valued Logics 6 (2001), 95–135.

    MathSciNet  MATH  Google Scholar 

  8. J. Hashimoto: On the product decomposition of partially ordered sets. Math. Jap. 1 (1948), 120–123.

    MathSciNet  MATH  Google Scholar 

  9. J. Jakubík: Direct product decompositions of partially ordered groups. Czechoslovak Math. J. 10 (1960), 231–243. (In Russian.)

    MathSciNet  Google Scholar 

  10. J. Jakubík: Cantor-Bernstein theorem for lattice ordered groups. Czechoslovak Math. J. 22 (1972), 159–175.

    MathSciNet  Google Scholar 

  11. J. Jakubík, M. Csontóová: Convex isomorphisms of directed multilattices. Math. Bohem. 118 (1993), 359–378.

    MathSciNet  MATH  Google Scholar 

  12. J. Jakubík: Complete lattice ordered groups with strong units. Czechoslovak Math. J. 46 (1996), 221–230.

    MathSciNet  MATH  Google Scholar 

  13. J. Jakubík: Convex isomorphisms of archimedean lattice ordered groups. Mathware and Soft Computing 5 (1998), 49–56.

    MathSciNet  MATH  Google Scholar 

  14. J. Jakubík: Cantor-Bernstein theorem for MV-algebras. Czechoslovak Math. J. 49 (1999), 517–526.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Jakubík: Direct product decompositions of infinitely distributive lattices. Math. Bohemica 125 (2000), 341–354.

    MATH  Google Scholar 

  16. J. Jakubík: Convex mappings of archimedean MV-algebras. Math. Slovaca 51 (2001), 383–391.

    MathSciNet  MATH  Google Scholar 

  17. J. Jakubík: Direct product decompositions of pseudo MV-algebras. Arch. Math. 37 (2002), 131–142.

    Google Scholar 

  18. J. Jakubík: Cantor-Bernstein theorem for lattices. Math. Bohem. 127 (2002), 463–471.

    MathSciNet  MATH  Google Scholar 

  19. J. Jakubík: A theorem of Cantor-Bernstein type for orthogonally σ-complete pseudo MV-algebras. Tatra Mt. Math. Publ. 22 (2002), 91–103.

    Google Scholar 

  20. G. Jenča: A Cantor-Bernstein type theorem for effect algebras. Algebra Univers. 48 (2002), 399–411.

    Article  MATH  Google Scholar 

  21. A. G. Kurosh: Group Theory. Nauka, Moskva, 1953. (In Russian.)

    Google Scholar 

  22. J. Rachůnek: A non-commutative generalization of MV-algebras. Czechoslovak Math. J. 52 (2002), 255–273.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Sikorski: A generalization of theorem of Banach and Cantor-Bernstein. Colloq. Math. 1 (1948), 140–144.

    MathSciNet  MATH  Google Scholar 

  24. A. Tarski: Cardinal Algebras. Oxford University Press, New York, 1949.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Matematický ústav SAV, Grešákova 6, 040 01, Košice, Slovakia

    Ján Jakubík

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  1. Ján Jakubík
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Correspondence to Ján Jakubík.

Additional information

This work was supported by Science and Technology Assistance Agency under the contract No APVT-51-032002.

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Jakubík, J. On a theorem of Cantor-Bernstein type for algebras. Czech Math J 58, 1–14 (2008). https://doi.org/10.1007/s10587-008-0001-3

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  • DOI: https://doi.org/10.1007/s10587-008-0001-3

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