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

Subject Areas : Transactions on Fuzzy Sets and Systems

Afrodita Iorgulescu ¹ (Department of Informatics and Economic Cybernetics, Bucharest University of Economic Studies Bucharest, Romania.)

Received: 2022-02-20 Accepted : 2022-02-20 Published : 2022-05-07

Keywords: 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,

Abstract :

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, ⊙, , 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.

References:

[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).