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lattices; central elements; factor congruences; varieties
The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to $\sigma $-complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for $\sigma $-complete orthomodular lattices, Stone algebras, $BL$-algebras, $MV$-algebras, pseudo $MV$-algebras, Łukasiewicz and Post algebras of order $n$.
[1] R. Balbes and Ph.  Dwinger: Distributive Lattices. University of Missouri Press, Columbia, 1974. MR 0373985
[2] G.  Birkhoff: Lattice Theory, Third Edition. AMS, Providence, 1967. MR 0227053
[3] V.  Boicescu, A,  Filipoiu, G.  Georgescu and S. Rudeanu: Łukasiewicz-Moisil Algebras. North-Holland, Amsterdam, 1991. MR 1112790
[4] R. Cignoli: Representation of Łukasiewicz and Post algebras by continuous functions. Colloq. Math. 24 (1972), 127–138. MR 0307996 | Zbl 0246.02043
[5] R. Cignoli: Lectures at Buenos Aires University. 2000.
[6] R.  Cignoli, M. I. D’Ottaviano and D.  Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht, 2000. MR 1786097
[7] R. Cignoli and A.  Torrens: An algebraic analysis of product logic. Multiple Valued Logic 5 (2000), 45–65. MR 1743553
[8] A. De Simone, D. Mundici and M.  Navara: A Cantor-Bernstein Theorem for complete $MV$-algebras. Czechoslovak Math.  J 53 (2003), 437–447. DOI 10.1023/A:1026299723322 | MR 1983464
[9] A. De Simone, M.  Navara and P.  Pták: On interval homogeneous orthomodular lattices. Comment. Math. Univ. Carolin. 42 (2001), 23–30. MR 1825370
[10] A. Dvurečenskij: Pseudo $MV$-algebras are intervals in $l$-groups. J.  Austral. Math. Soc. (Ser.  A) 72 (2002), 427–445. DOI 10.1017/S1446788700036806
[11] A.  Dvurečenskij: On pseudo $MV$-algebras. Soft Computing 5 (2001), 347–354. DOI 10.1007/s005000100136
[12] G.  Georgescu and A. Iorgulescu: Pseudo $MV$-algebras. 6 (2001), 95–135. MR 1817439
[13] P. Hájek: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998. MR 1900263
[14] U.  Höhle: Commutative, residuated $l$-monoids. In: Non-Classical Logics and their Applications to Fuzzy Subset. A Handbook on the Mathematical Foundations of Fuzzy Set Theory, U.  Höhle, E. P. Klement (eds.), Kluwer, Dordrecht, 1995. MR 1345641
[15] J.  Jakubík: A theorem of Cantor-Bernstein type for orthogonally $\sigma $-complete pseudo $MV$-algebras. Czechoslovak Math.  J (to appear). MR 1889037
[16] J. A.  Kalman: Lattices with involution. Trans. Amer. Math. Soc. 87 (1958), 485–491. DOI 10.1090/S0002-9947-1958-0095135-X | MR 0095135 | Zbl 0228.06003
[17] S.  Koppelberg: Handbook of Boolean Algebras, Vol. 1 (J. Donald Monk, ed.). North Holland, Amsterdam, 1989. MR 0991565
[18] T.  Kowalski and H.  Ono: Residuated Lattices: An algebraic glimpse at logics without contraction. Preliminary report, 2000.
[19] F.  Maeda and S.  Maeda: Theory of Symmetric Lattices. Springer-Verlag, Berlin, 1970. MR 0282889
[20] R.  Sikorski: A generalization of theorem of Banach and Cantor-Bernstein. Colloq. Math. 1 (1948), 140–144. MR 0027264
[21] A. Tarski: Cardinal Algebras. Oxford University Press, New York, 1949. MR 0029954 | Zbl 0041.34502
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