Abstract
We study the structures which satisfy a generalization of the Cantor–Bernstein theorem. This work is inspired by related results concerning quantum structures (orthomodular lattices). It has been proved that σ-complete MV-algebras satisfy a version of the Cantor–Bernstein theorem which assumes that the bounds of isomorphic intervals are boolean. This result has been extended to more general structures, e.g., effect algebras and pseudo-BCK-algebras.
There is another direction of research which has been paid less attention. We ask which algebras satisfy the Cantor–Bernstein theorem in the same form as for σ-complete boolean algebras (due to Sikorski and Tarski) without any additional assumption. In the case of orthomodular lattices, it has been proved that this class is rather large. E.g., every orthomodular lattice can be embedded as a subalgebra or expressed as an epimorphic image of a member of this class. On the other hand, also the complement of this class is large in the same sense. We study the analogous question for MV-algebras and we find out interesting examples of MV-algebras which possess or do not possess this property. This contributes to the mathematical foundations by showing the scope of validity of the Cantor–Bernstein theorem in its original form.
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References
Beran, L.: Orthomodular Lattices. Algebraic Approach. D. Reidel, Dordrecht (1984)
Cignoli, R., Mundici, D.: An invitation to Chang’s MV-algebras. In: Droste, M., Göbel, R. (eds.) Advances in Algebra and Model Theory, pp. 171–197. Gordon and Breach Publishing Group, Reading (1997)
Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Trends in Logic, vol. 7. Kluwer Academic Publishers, Dordrecht (1999)
De Simone, A., Mundici, D., Navara, M.: A Cantor–Bernstein theorem for σ-complete MV-algebras. Czechoslovak Math. J. 53(128) No. 2, 437–447 (2003)
De Simone, A., Navara, M., Pták, P.: On interval homogeneous orthomodular lattices. Comment. Math. Univ. Carolin. 42(1), 23–30 (2001)
De Simone, A., Navara, M.: On the permanence properties of interval homogeneous orthomodular lattices. Math. Slovaca 54, 13–21 (2004)
Di Nola, A., Navara, M.: Cantor–Bernstein property for MV-algebras. To appear
Dvurečenskij, A.: Central elements and Cantor–Bernstein theorem for pseudo-effect algebras. J. Austral. Math. Soc. 74, 121–143 (2003)
Freytes, H.: An algebraic version of the Cantor–Bernstein–Schröder Theorem. Czechoslovak J. Math. 54, 609–621 (2004)
Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. AMS, Providence (1986)
Grätzer, G.: General Lattice Theory, 2nd edn. J. Wiley, Basel (1998)
Hanf, W.: On some fundamental problems concerning isomorphism of boolean algebras. Math. Scand. 5, 205–217 (1957)
J., J.: Cantor–Bernstein theorem for MV-algebras. Czechoslovak Math. J. 49(124), 517–526 (1999)
Jakubík, J.: On orthogonally σ-complete lattice ordered groups. Czechoslovak Math. J. 52(128) No. 4, 881–888 (2002)
Jakubík, J.: A theorem of Cantor–Bernstein type for orthogonally σ-complete pseudo MV-algebras. Tatra Mt. Math. Publ. 22, 91–103 (2002)
Jakubík, J.: Convex mappings of archimedean MV-algebras. Math. Slovaca, to appear
Jakubík, J.: Cantor–Bernstein theorem for lattices. Submitted
Jenča, G.: A Cantor–Bernstein type theorem for effect algebras. Algebra Univers. 48, 399–411 (2002)
Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)
Kallus, M., Trnková, V.: Symmetries and retracts of quantum logics. Int. J. Theor. Phys. 26, 1–9 (1987)
Kinoshita, S.: A solution to a problem of Sikorski. Fund. Math. 40, 39–41 (1953)
Kühr, J.: Cantor–Bernstein theorem for pseudo-BCK-algebras. Preprint (2006)
Levy, A.: Basic Set Theory. Perspectives in Mathematical Logic. Springer, Berlin (1979)
Monk, J.D., Bonnet, R.: Handbook of Boolean Algebras I. North-Holland, Amsterdam (1989)
Mundici, D.: Interpretation of AF C ∗ -algebras in Łukasiewicz sentential calculus. J. Functional Analysis 65, 15–63 (1986)
Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991)
Riečan, B., Mundici, D.: Probability on MV-algebras. In: Pap, E. (ed.) Handbook of Measure Theory, North-Holland, Amsterdam (2001)
Sikorski, R.: Boolean Algebras. Ergebnisse Math. Grenzgeb. Springer, Berlin (1960)
Sikorski, R.: A generalization of a theorem of Banach and Cantor–Bernstein. Colloquium Math. 1, 140–144 and 242 (1948)
Tarski, A.: Cardinal Algebras. Oxford University Press, New York (1949)
Trnková, V.: Automorphisms and symmetries of quantum logics. Int. J. Theor. Physics 28, 1195–1214 (1989)
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Di Nola, A., Navara, M. (2007). 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. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72434-6_87
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DOI: https://doi.org/10.1007/978-3-540-72434-6_87
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