Abstract
We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers.
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References
Aglianò, P., and A. Ursini, ‘On subtractive varieties II: General properties’, Algebra Universalis, 36 (1996), 222–259.
Cattaneo, G., M. L. Dalla Chiara, R. Giuntini, and R. Leporini, ‘An unsharp logic from quantum computation’, International Journal of Theoretical Physics, 43, 7–8 (2004), 1803–1817.
Cattaneo, G., M. L. Dalla Chiara, R. Giuntini, and R. Leporini, ‘Quantum computational structures’, Mathematica Slovaca, 54 (2004), 87–108.
Chang, C. C., ‘A new proof of the completeness of Lukasiewicz axioms’, Transactions of the American Mathematical Society, 93 (1959), 74–90.
Cignoli, R., I. M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer, Dordrecht, 1999.
Dalla Chiara, M. L., R. Giuntini, and R. Greechie, Reasoning in Quantum Theory, Kluwer, Dordrecht, 2004.
Dalla Chiara, M. L., R. Giuntini, and R. Leporini, ‘Quantum computational logics: A survey’, in V. F. Hendricks, J. Malinowski (eds.), Trends in Logic: 50 Years of Studia Logica, Kluwer, Dordrecht, 2003, pp. 213–255.
Gumm, H. P., and A. Ursini, ‘Ideals in universal algebra’, Algebra Universalis, 19 (1984), 45–54.
Hagermann, J., and C. Hermann, ‘A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity’, Archive for Mathematics, 32 (1979), 234–245.
Idziak, P. M., ‘Lattice operations in BCK algebras’, Mathematica Japonica, 29, 6 (1984), 839–846.
Maltsev, A. I., ‘On the general theory of algebraic systems’(in Russian), Mat. Sb. (N. S.), 35, 77 (1954), 3–20.
Mitschke, A., ‘Implication algebras are 3-permutable and 3-distributive’, Algebra Universalis, 1 (1971), 182–186.
Paoli, F., Substructural Logics: A Primer, Kluwer, Dordrecht, 2002.
Ursini, A., ‘On subtractive varieties I’, Algebra Universalis, 31 (1994), 204–222.
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Ledda, A., Konig, M., Paoli, F. et al. MV-Algebras and Quantum Computation. Stud Logica 82, 245–270 (2006). https://doi.org/10.1007/s11225-006-7202-2
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DOI: https://doi.org/10.1007/s11225-006-7202-2