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Cantor-Bernstein theorem; MV-algebra; boolean element of an MV-algebra; partition of unity; direct product decomposition; $\sigma $-complete MV-algebra
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.
[1] R.  Cignoli and D. Mundici: 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. MR 1683528
[2] R.  Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-valued Reasoning. Trends in Logic. Vol.  7. Kluwer Academic Publishers, Dordrecht, 1999. MR 1786097
[3] W. Hanf: On some fundamental problems concerning isomorphism of boolean algebras. Math. Scand. 5 (1957), 205–217. MR 0108451 | Zbl 0081.26101
[4] J. Jakubík: Cantor-Bernstein theorem for $MV$-algebras. Czechoslovak Math. J. 49(124) (1999), 517–526. DOI 10.1023/A:1022467218309 | MR 1708370
[5] S.  Kinoshita: A solution to a problem of Sikorski. Fund. Math. 40 (1953), 39–41. MR 0060809
[6] A.  Levy: Basic Set Theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1979. MR 0533962
[7] D.  Mundici: Interpretation of AF $C^{*}$-algebras in Łukasiewicz sentential calculus. J.  Funct. Anal. 65 (1986), 15–63. DOI 10.1016/0022-1236(86)90015-7 | MR 0819173 | Zbl 0597.46059
[8] R.  Sikorski: Boolean Algebras. Springer-Verlag. Ergebnisse Math. Grenzgeb., Berlin, 1960. MR 0126393 | Zbl 0087.02503
[9] R. Sikorski: A generalization of a theorem of Banach and Cantor-Bernstein. Colloq. Math. 1 (1948), 140–144 and 242. MR 0027264
[10] A.  Tarski: Cardinal Algebras. Oxford University Press, New York, 1949. MR 0029954 | Zbl 0041.34502
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