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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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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