On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices

Transactions on Fuzzy Sets and Systems

مقاله 2، دوره 1، شماره 1، مرداد 2022، صفحه 1-41   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/tfss.2022.690286

نویسنده

Afrodita Iorgulescu^*

Department of Informatics and Economic Cybernetics‎, Bucharest University of Economic Studies, Bucharest‎, ‎Romania‎

چکیده

‎Orthomodular lattices generalize the Boolean algebras; they have arisen‎ ‎in the study of quantum logic‎. ‎Quantum-MV algebras were introduced‎ ‎as non-lattice theoretic generalizations of MV algebras and as non-idempotent generalizations of orthomodular lattices‎.
‎In this paper‎, ‎we continue the research in the ``world'' of involutive  algebras of the form‎ ‎$(A‎, ‎\odot‎, ‎{}^-,1)$‎, ‎with $1^-=0 $‎, $1$ being the last element‎. ‎We clarify now some aspects concerning the quantum-MV (QMV) algebras as non-idempotent generalizations of orthomodular lattices‎.
‎We study in some detail the orthomodular lattices (OMLs)‎ ‎and we introduce and study two generalizations of them‎, ‎the orthomodular softlattices (OMSLs) and the orthomodular widelattices (OMWLs)‎. ‎We establish systematically connections between‎ ‎OMLs and OMSLs/OMWLs and QMV‎, ‎pre-MV‎, ‎metha-MV‎, ‎orthomodular algebras and ortholattices‎, ‎orthosoftlattices/orthowidelattices‎ - ‎connections illustrated in 22 Figures‎. ‎We prove‎, ‎among others‎, ‎that the transitive OMLs coincide with the Boolean algebras‎, ‎that the OMSLs coincide with the OMLs‎, ‎that the OMLs are included in OMWLs and that the OMWLs are a proper subclass of QMV algebras‎. ‎The transitive and/or the antisymmetric case is also studied‎.

کلیدواژه‌ها

m-MEL algebra؛ m-BE algebra؛ m-pre-BCK algebra؛ m-BCK algebra؛ MV algebra؛ Quantum-MV algebra؛ Pre-MV algebra؛ Metha-MV algebra؛ Orthomodular algebra؛ Ortholattice؛ Orthosoftlattice؛ Orthowidelattice؛ Boolean algebra

مراجع

[1] G. Birkhoff, Lattice Theory, Colloquium Publications 25, American Mathematical Society, Providence Rhode Island, (1967).
[2] G. Birkhoff and J. Von Neumann, The logic of quantum mechanics, Ann. Math., 37 (1936), 823-843.
[3] Ch. Ch. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc., 88 (1958), 467-490.
[4] M. L. D. Chiara, R. Giuntini and R. Greechie, Reasoning in Quantum Theory, Sharp and Unsharp Quantum Logics, Springer, Switzerland, (2004).
[5] R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic Foundations of Many-valued Reasoning, Kluwer Academic Publishers & Springer Science, Dordrecht, (2000).
[6] A. Dvurecenskij and S. Pulmannová, New Trends in Quantum Structures, Mathematics and Its Applications, Kluwer Academic Publishers & Springer Science, Dordrecht, (2000).
[7] R. Giuntini, An independent axiomatization of quantum MV algebras. In: C. Carola, A. Rossi (eds.), The Foundations of Quantum Mechanics, World Scientific, Singapore, (2000).
[8] R. Giuntini, Quantum Logic and Hidden Variables, B.I. Wissenschaftfsverlag, Mannheim, Wien, Zürich, (1991).
[9] R. Giuntini, Quantum MV-algebras, Studia Logica, 56 (1996), 393-417.
[10] R. Giuntini, Quantum MV-algebras and commutativity, Inter. J. Theor. Phys., 37 (1998), 65-74.
[11] R. Giuntini, Quasilinear QMV algebras, Inter. J. Theor. Phys., 34 (1995), 1397-1407.
[12] R. Giuntini, Unsharp orthoalgebras and quantum MV algebras, In: The Foundations of Quantum Mechanics, Kluwer Academic Publishers & Springer Science, Dordrecht, (1996).
[13] R. Giuntini, Weakly linear quantum MV-algebras, Algebra Universalis, 53 (2005), 45-72.
[14] A. Iorgulescu, Algebras of logic as BCK algebras, Academy of Economic Studies Press, Bucharest, (2008).
[15] A. Iorgulescu, Algebras of logic vs. Algebras, Landscapes in Logic, Vol. 1 Contemporary Logic and Computing, Editor Adrian Rezus, College Publications, United Kingdom, (2020).
[16] A. Iorgulescu, Implicative-groups vs. groups and generalizations, Matrix Rom, Bucharest, (2018).
[17] A. Iorgulescu, New generalizations of BCI, BCK and Hilbert algebras, Parts I and II (Dedicated to Dragos Vaida), J. Mult.-Valued Logic Soft Comput., 4 (2016), 353-406, and 407-456. Available online: http://arxiv.org/abs/1312.2494
[18] A. Iorgulescu, On Quantum-MV algebras-Part I: Orthomodular algebras, Sci. Ann. Comput. Sci., 31(2) (2021), 163-221.
[19] A. Iorgulescu, On Quantum-MV algebras-Part III: The properties (m-Pabs-i) and (WNMm), To appear.

[20] A. Iorgulescu and M. Kinyon, Putting bounded involutive lattices, De Morgan algebras, ortholattices and Boolean algebras on the “map”, Journal of Applied Logics (JALs-ifCoLog), Special Issue on Multiple Valued Logic and Applications, 8(5) (2021), 1169-1213.
[21] A. Iorgulescu and M. Kinyon, Putting Quantum MV algebras on the “map”, Sci. Math. Jpn., (2021). Availablle online: https://www.jams.or.jp/notice/scmjol/2021.html.
[22] A. Iorgulescu and M. Kinyon, Two generalizations of bounded involutive lattices and of ortholattices, J. Appl. Log. (JALs-ifCoLog), 8(7) (2021), 2173-2218.
[23] G. Kalmbach, Orthomodular Lattices, Academic Press, London, New York, etc., (1983).
[24] W. W. Mc. Cune, Prover9 and Mace4 program. Available online: http://www.cs.unm.edu/ mccune/Prover9.
[25] R. Padmanabhan and S. Rudeanu, Axioms for Lattices and Boolean Algebras, World Scientific, Singapore, etc., (2008).