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The Cantor–Bernstein–Schröder theorem via universal algebra

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Abstract

The Cantor–Bernstein–Schröder theorem (CBS-theorem for short) of set theory was generalized by Sikorski and Tarski to \(\sigma \)-complete Boolean algebras. After this, several generalizations of the CBS-theorem, extending the Sikorski–Tarski version to different classes of algebras, have been established. Among these classes there are lattice ordered groups, orthomodular lattices, MV-algebras, residuated lattices, etc. This suggests to consider a common algebraic framework in which the algebraic versions of the CBS-theorem can be formulated. In this work we provide this framework establishing necessary and sufficient conditions for the validity of the theorem. We also show how this abstract framework includes the versions of the CBS-theorem already present in the literature as well as new versions of the theorem extended to other classes such as groups, modules, semigroups, rings, \(*\)-rings etc.

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References

  1. Balbes, R., Dwinger, Ph: Distributive Lattices. University of Missouri Press, Columbia (1974)

    MATH  Google Scholar 

  2. Berberian, S.K.: Baer Rings and Baer \(^*\)-Rings. University of Texas, Austin (2003). (preprint)

    MATH  Google Scholar 

  3. Berberian, S.K.: Baer \(^*\)-Rings. Springer, New York (2011)

    MATH  Google Scholar 

  4. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  5. Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S.: Łukasiewicz-Moisil Algebras. North-Holland, Amsterdam (1991)

    MATH  Google Scholar 

  6. Borel, E.: Leçons sur la Théorie des Fonctions. Gauthier-Villars et fils, Paris (1898)

    MATH  Google Scholar 

  7. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1982)

    MATH  Google Scholar 

  8. Cignoli, R.: Representation of Łukasiewicz and Post algebras by continuous functions. Colloq. Math. 24, 128–138 (1972)

    Article  Google Scholar 

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

    Book  Google Scholar 

  10. Cignoli, R., Torrens, A.: An algebraic analysis of product logic. Mult. Valued Log. 5, 45–65 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Cintula, P.: About axiomatic systems of product fuzzy logic. Soft Comput. 5, 243–244 (2010)

    Article  Google Scholar 

  12. De Simone, A., Mundici, D., Navara, M.: A Cantor–Bernstein theorem for \(\sigma \)-complete MV-algebras. Czechoslov. Math. J. 53, 437–447 (2003)

    Article  MathSciNet  Google Scholar 

  13. De Simone, A., Navara, M., Pták, P.: On interval homogeneous orthomodular lattices. Comment. Math. Univ. Carolin. 42, 23–30 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Di Nola, A., Navara, M.: MV-Algebras with the Cantor–Bernstein property. In: Castillo, O., Melin, P., Ross, O.M., Sepúlveda, Cruz R., Pedrycz, W., Kacprzyk, J. (eds.) Theoretical Advances and Applications of Fuzzy Logic and Soft Computing. Advances in Soft Computing, vol. 42, pp. 861–868. Springer, Berlin (2007)

    Google Scholar 

  15. Dvurečenskij, A.: Pseudo MV-algebras are intervals in l-groups. J. Austral. Math. Soc. 72, 427–445 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Emanovský, P., Kühr, J.: Some properties of pseudo-BCK- and pseudo-BCI-algebras. Fuzzy Sets Syst. 339, 1–16 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Georgescu G., Iorgulescu A.: Pseudo-BCK algebras: an extension of BCK-algebra. In: Proceedings of the DMTCS’01, Combinatoric, Computability and Logic, pp. 97–114. London (2001)

  20. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)

    Book  Google Scholar 

  21. Iskander, A.: Factorable congruences and strict refinement. Acta Math. Univ. Comenianae 65, 101–109 (1996)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Jakubík, J.: A theorem of Cantor-Bernstein type for orthogonal \(\sigma \)-complete pseudo MV-algebras. Tatra Mt. Math. Publ. 22, 91–103 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Jakubík, J.: On orthogonal \(\sigma \)-complete lattice ordered groups. Czechoslov. Math. J. 52, 881–888 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, Jorge (ed.) Ordered Algebraic Structures, Developments in Mathematics, pp. 19–56. Springer, Boston (2002)

    Chapter  Google Scholar 

  27. Kalman, J.: Lattices with involution. Trans. Am. Math. Soc. 87, 485–491 (1958)

    Article  MathSciNet  Google Scholar 

  28. Kaplansky, I.: Rings of Operators, 2nd edn. Benjamin, New York (1968)

    MATH  Google Scholar 

  29. Korselt, A.: Über einen Beweis des Äquivalenzsatzes. Math. Ann. 70, 294–296 (1911)

    Article  MathSciNet  Google Scholar 

  30. Kowalski T., Ono H.: Residuated lattices: an algebraic glimpse at logics without contraction. Preliminary report (2000)

  31. Kühr, J.: Cantor Bernstein theorem for pseudo-BCK-algebras. Int. J. Theor. Phys. 47, 212–222 (2008)

    Article  MathSciNet  Google Scholar 

  32. Maeda, F., Maeda, S.: Theory of Symmetric Lattices. Springer, Berlin (1970)

    Book  Google Scholar 

  33. Manzonetto G., Salibra A.: From \(\lambda \)-calculus to universal algebra and back. In: Proceedings of the 33rd International Symposium on Mathematical Foundations of Computer Science, LNCS, vol. 5162, pp. 479–490. Springer (2008)

  34. Monk, J.: On pseudo-simple universal algebras. Proc. Am. Math. Soc. 13, 543–546 (1962)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Schröder, E., Über, G.: Cantor’sche Sätze. Jahresbericht der Deutschen Mathematiker-Vereinigung 5, 81–82 (1896)

    MATH  Google Scholar 

  37. Schröder, E.: Ueber zwei der Endlichkeit und G. Cantor’sche Sätze. Nova Acta Academiae Caesareae Leopoldino-Carolineae (Halle a. d. Saale) 71, 365–376 (1898)

    MATH  Google Scholar 

  38. Steinberg, S.: Lattice-Ordered Rings and Modules. Springer, New York (2010)

    Book  Google Scholar 

  39. Swamy, U.M., Murti, G.S.: Boolean centre of a universal algebra. Algebra Univ. 13, 202–205 (1981)

    Article  MathSciNet  Google Scholar 

  40. Swamy, U.M., Murti, G.S.: Boolean centre of a semigroup. Pure Appl. Math. Sci. 13, 1–2 (1981)

    MathSciNet  MATH  Google Scholar 

  41. Tarski, A.: Cardinal Algebras. Oxford University Press, New York (1949)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Philosophy/Mathematics, University of Cagliari, Via Is Mirionis I, 09123, Cagliari, Italy

    Hector Freytes

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  1. Hector Freytes
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Correspondence to Hector Freytes.

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In memoriam: Roberto Cignoli (1937–2018).

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This research was partly supported by Horizon 2020 program of the European Commission: SYSMICS project, number: 689176, MSCA-RISE-2015 and Fondazione Banco di Sardegna project “Science and its Logics”, Cagliari, number: F72F16003220002”. The author expresses his gratitude to Giuseppe Sergioli for his careful reading of this work and to the anonymous referee for his extremely accurate comments.

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Freytes, H. The Cantor–Bernstein–Schröder theorem via universal algebra. Algebra Univers. 80, 17 (2019). https://doi.org/10.1007/s00012-019-0590-8

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