Algebra universalis

, 80:17 | Cite as

The Cantor–Bernstein–Schröder theorem via universal algebra

  • Hector FreytesEmail author


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.


Cantor–Bernstein–Schröder theorem Congruence lattice Factor congruences 

Mathematics Subject Classification

03G10 06C15 



  1. 1.
    Balbes, R., Dwinger, Ph: Distributive Lattices. University of Missouri Press, Columbia (1974)zbMATHGoogle Scholar
  2. 2.
    Berberian, S.K.: Baer Rings and Baer \(^*\)-Rings. University of Texas, Austin (2003). (preprint) zbMATHGoogle Scholar
  3. 3.
    Berberian, S.K.: Baer \(^*\)-Rings. Springer, New York (2011)zbMATHGoogle Scholar
  4. 4.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)zbMATHGoogle Scholar
  5. 5.
    Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S.: Łukasiewicz-Moisil Algebras. North-Holland, Amsterdam (1991)zbMATHGoogle Scholar
  6. 6.
    Borel, E.: Leçons sur la Théorie des Fonctions. Gauthier-Villars et fils, Paris (1898)zbMATHGoogle Scholar
  7. 7.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1982)zbMATHGoogle Scholar
  8. 8.
    Cignoli, R.: Representation of Łukasiewicz and Post algebras by continuous functions. Colloq. Math. 24, 128–138 (1972)CrossRefGoogle Scholar
  9. 9.
    Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  10. 10.
    Cignoli, R., Torrens, A.: An algebraic analysis of product logic. Mult. Valued Log. 5, 45–65 (2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cintula, P.: About axiomatic systems of product fuzzy logic. Soft Comput. 5, 243–244 (2010)CrossRefGoogle Scholar
  12. 12.
    De Simone, A., Mundici, D., Navara, M.: A Cantor–Bernstein theorem for \(\sigma \)-complete MV-algebras. Czechoslov. Math. J. 53, 437–447 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    De Simone, A., Navara, M., Pták, P.: On interval homogeneous orthomodular lattices. Comment. Math. Univ. Carolin. 42, 23–30 (2001)MathSciNetzbMATHGoogle Scholar
  14. 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. 15.
    Dvurečenskij, A.: Pseudo MV-algebras are intervals in l-groups. J. Austral. Math. Soc. 72, 427–445 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dvurečenskij, A.: Central elements and Cantor–Bernstein s theorem for pseudo-effect algebras. J. Aust. Math. Soc. 74, 121–143 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Emanovský, P., Kühr, J.: Some properties of pseudo-BCK- and pseudo-BCI-algebras. Fuzzy Sets Syst. 339, 1–16 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Freytes, H.: An algebraic version of the Cantor–Bernstein–Schröder theorem. Czechoslov. Math. J. 54, 609–621 (2004)MathSciNetCrossRefGoogle Scholar
  19. 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)Google Scholar
  20. 20.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
  21. 21.
    Iskander, A.: Factorable congruences and strict refinement. Acta Math. Univ. Comenianae 65, 101–109 (1996)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Jakubík, J.: Cantor-Bernstein theorem for MV-algebras. Czechoslov. Math. J. 49, 517–526 (1999)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jakubík, J.: A theorem of Cantor-Bernstein type for orthogonal \(\sigma \)-complete pseudo MV-algebras. Tatra Mt. Math. Publ. 22, 91–103 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Jakubík, J.: On orthogonal \(\sigma \)-complete lattice ordered groups. Czechoslov. Math. J. 52, 881–888 (2002)CrossRefGoogle Scholar
  25. 25.
    Jenča, G.: A Cantor Bernstein type theorem for effect algebras. Algebra Univ. 48, 399–411 (2002)MathSciNetCrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 27.
    Kalman, J.: Lattices with involution. Trans. Am. Math. Soc. 87, 485–491 (1958)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kaplansky, I.: Rings of Operators, 2nd edn. Benjamin, New York (1968)zbMATHGoogle Scholar
  29. 29.
    Korselt, A.: Über einen Beweis des Äquivalenzsatzes. Math. Ann. 70, 294–296 (1911)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kowalski T., Ono H.: Residuated lattices: an algebraic glimpse at logics without contraction. Preliminary report (2000)Google Scholar
  31. 31.
    Kühr, J.: Cantor Bernstein theorem for pseudo-BCK-algebras. Int. J. Theor. Phys. 47, 212–222 (2008)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Maeda, F., Maeda, S.: Theory of Symmetric Lattices. Springer, Berlin (1970)CrossRefGoogle Scholar
  33. 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)Google Scholar
  34. 34.
    Monk, J.: On pseudo-simple universal algebras. Proc. Am. Math. Soc. 13, 543–546 (1962)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sikorski, R.: A generalization of theorems of Banach and Cantor–Bernstein. Colloq. Math. 1, 140–144 (1948)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Schröder, E., Über, G.: Cantor’sche Sätze. Jahresbericht der Deutschen Mathematiker-Vereinigung 5, 81–82 (1896)zbMATHGoogle Scholar
  37. 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)zbMATHGoogle Scholar
  38. 38.
    Steinberg, S.: Lattice-Ordered Rings and Modules. Springer, New York (2010)CrossRefGoogle Scholar
  39. 39.
    Swamy, U.M., Murti, G.S.: Boolean centre of a universal algebra. Algebra Univ. 13, 202–205 (1981)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Swamy, U.M., Murti, G.S.: Boolean centre of a semigroup. Pure Appl. Math. Sci. 13, 1–2 (1981)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Tarski, A.: Cardinal Algebras. Oxford University Press, New York (1949)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Philosophy/MathematicsUniversity of CagliariCagliariItaly

Personalised recommendations