Abstract
The purpose of the present article is to extend the scope of some investigations about abstract logics arising quite naturally out of Quasi-MV algebras (for short, qMV algebras) also to \(\sqrt{^{\prime}}\) qMV algebras. We will therefore introduce, mutually compare and (in some cases) axiomatise several logics arising out of the variety of \(\sqrt{^{\prime}}\) qMV algebras and out of some important subclasses of such. Subsequently, we will investigate the same logics by resorting to the methods and techniques of abstract algebraic logic.
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Aglianò, P., Ursini, A.: On subtractive varieties III: from ideals to congruences. Algebra Univers. 37, 296–333 (1997)
Aharonov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. In: Proc. 13th Annual ACM Symp. on Theory of Computation, STOC, pp. 20–30 (1997)
Berman, J., Blok, W.J.: Algebras defined from ordered sets and the varieties they generate. Order 23(1), 65–88 (2006)
Blok, W.J., Pigozzi, D.: Algebraizable logics. Mem. Am. Math. Soc. 77(396) (1989)
Bou, F., Paoli, F., Ledda, A., Freytes, H.: On some properties of quasi-MV algebras and \(\sqrt{^{\prime}}\) quasi-MV algebras. Part II. Soft Comput. 12(4), 341–352 (2008)
Bou, F., Paoli, F., Ledda, A., Spinks, M., Giuntini, R.: The logic of quasi-MV algebras. J. Log. Comput. 20(2), 619–643 (2010)
Chajda, I.: Normally presented varieties. Algebra Univers. 34, 327–335 (1995)
Czelakowski, J., Jansana, R.: Weakly algebraizable logics. J. Symb. Log. 65, 641–668 (2000)
Czelakowski, J., Pigozzi, D.: Fregean logics. Ann. Pure Appl. Log. 127, 17–76 (2004)
Dalla, Chiara M.L., Giuntini, R., Leporini, R.: Logics from quantum computation. Int. J. Quantum Inf. 3(2), 293–337 (2005)
Duan, R., Ji, Z., Feng, Y., Ying, M.: Quantum operation, quantum Fourier transform and semi-definite programming. Phys. Lett. A 323, 48–56 (2004)
Gudder, S.: Quantum computational logic. Int. J. Theor. Phys. 42, 39–47 (2003)
Font, J.M.: Taking degrees of truth seriously. Stud. Log. 91, 383–406 (2009)
Font, J.M., Gil, A.J., Torrens, A., Verdù, V.: On the infinite-valued Lukasiewicz logic that preserves degrees of truth. Arch. Math. Log. 45, 839–868 (2006)
Font, J.M., Jansana, R.: A General Algebraic Semantics for Sentential Logics. Springer, Berlin (1996)
Font, J.M., Rodriguez, A.J., Torrens, A.: Wajsberg algebras. Stochastica 8, 5–31 (1984)
Giuntini, R., Freytes, H., Ledda, A., Paoli, F.: A discriminator variety of Gödel algebras with operators arising in quantum computation. Fuzzy Sets Syst. 160(8), 1082–1098 (2009)
Giuntini, R., Ledda, A., Paoli, F.: Expanding quasi-MV algebras by a quantum operator. Stud. Log. 87(1), 99–128 (2007)
Giuntini, R., Ledda, A., Paoli, F.: Categorical equivalences for \(\sqrt{^{\prime}}\) quasi-MV algebras. J. Log. Comput. 20(4), 795–810 (2010)
Kowalski, T., Paoli, F.: On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part III. Rep. Math. Log. 45, 161–199 (2010)
Kowalski, T., Paoli, F., Spinks, M.: Quasi-subtractive varieties. J. Symb. Log. (2011, forthcoming)
Ledda, A., Konig, M., Paoli, F., Giuntini, R.: MV algebras and quantum computation. Stud. Log. 82(2), 245–270 (2006)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Paoli, F., Ledda, A., Giuntini, R., Freytes, H.: On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part I. Rep. Math. Log. 44, 31–63 (2009)
Tarasov, V.: Quantum computer with mixed states and four-valued logic. J. Phys. A 35, 5207–5235 (2002)
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Paoli, F., Ledda, A., Spinks, M. et al. Logics from \(\sqrt{^{\prime}}\) Quasi-MV Algebras. Int J Theor Phys 50, 3882–3902 (2011). https://doi.org/10.1007/s10773-011-0865-0
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DOI: https://doi.org/10.1007/s10773-011-0865-0