SACS | On Quantum-MV Algebras – Part I: The Orthomodular Algebras

Published in Volume XXXI, Issue 2, 2021, pages 163-222, doi: 10.7561/SACS.2021.2.163

Authors: A. Iorgulescu

Abstract

We prove that almost all the properties of quantum-MV algebras are verified by orthomodular algebras, the new algebras introduced in a previous paper. We put a special insight on transitive antisymmetric orthomodular (taOM) algebras, generalizations of MV algebras. We make the connection with IMTL and NM algebras.

In memoriam Dragos. Vaida (1933 – 2020)

Full Text (PDF)

References

[1] Garrett Birkhoff. Lattice Theory, volume 25 of Colloquium Publications. American Mathematical Society, 3rd edition, 1967.

[2] Chen Chung Chang. Algebraic analysis of many valued logics. Transactions of the American Mathematical Society, 88(2):467-490, 1958. doi:10.1090/S0002-9947-1958-0094302-9.

[3] Maria Luisa Dalla Chiara, Roberto Giuntini, and Richard Greechie. Reasoning in Quantum Theory – Sharp and Unsharp Quantum Logics, volume 22 of Trends in Logic. Springer, Dordrecht, 2004. doi:10.1007/978-94-017-0526-4.

[4] Roberto L. O. Cignoli, Itala M. L. D’Ottaviano, and Daniele Mundici. Algebraic Foundations of Many-Valued Reasoning, volume 7 of Trends in Logic. Springer, Dordrecht, 2000. doi:10.1007/978-94-015-9480-6.

[5] Anatolij Dvurecenskij and Sylvia Pulmannova. New Trends in Quantum Structures, volume 516 of Mathematics and Its Applications. Kluwer Academic Publishers, 2000. doi:10.1007/978-94-017-2422-7.

[6] Francesc Esteva and Lluis Godo. Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124(3):271-288, 2001. doi:10.1016/S0165-0114(01)00098-7.

[7] Francesc Esteva, Lluis Godo, and Carles Noguera. On rational weak nilpotent minimum logics. Journal of Multiple-Valued Logic and Soft Computing, 12(1-2):9-32, 2006.

[8] Janos C. Fodor. Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems, 69(2):141-156, 1995. doi:10.1016/0165-0114(94)00210-X.

[9] Roberto Giuntini. Quantum Logic and Hidden Variables. B.I. Wissenschaftsverlag, 1991.

[10] Roberto Giuntini. Quasilinear QMV algebras. International Journal of Theoretical Physics, 34:1397-1407, 1995. doi:10.1007/BF00676251.

[11] Roberto Giuntini. Unsharp orthoalgebras and quantum MV algebras. In Claudio Garola and Arcangelo Rossi, editors, The Foundations of Quantum Mechanics | Historical Analysis and Open Questions. Fundamental Theories of Physics (An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application), volume 71, pages 325-337. Springer, 1995. doi:10.1007/978-94-011-0029-8_27.

[12] Roberto Giuntini. Quantum MV algebras. Studia Logica, 56(3):393-417, 1996. doi:10.1007/BF00372773.

[13] Roberto Giuntini. Quantum MV-algebras and commutativity. International Journal of Theoretical Physics, 37:65-74, 1998. doi:10.1023/A:1026609121718.

[14] Roberto Giuntini. An independent axiomatization of quantum MV algebras. In Claudio Garola and Arcangelo Rossi, editors, The Foundations of Quantum Mechanics, pages 233-249. World Scientific, 2000. doi:10.1142/9789812793560_0017.

[15] Roberto Giuntini. Weakly linear quantum MV-algebras. Algebra Universalis, 53:45-72, 2005. doi:10.1007/s00012-005-1907-3.

[16] Stanley Gudder. Total extension of effect algebras. Foundations of Physics Letters, 8:243-252, 1995. doi:10.1007/BF02187348.

[17] Afrodita Iorgulescu. Algebras of logic as BCK algebras. Academy of Economic Studies Press, Bucharest, 2008.

[18] Afrodita Iorgulescu. Quasi-algebras versus regular algebras – Part I. Scientific Annals of Computer Science, 25(1):89{131, 2015. doi:10.7561/SACS.2015.1.89.

[19] Afrodita Iorgulescu. New generalizations of BCI, BCK and Hilbert algebras – Parts I and II. Journal of Multiple-Valued Logic and Soft Computing, 27(4):353-406 and 407-456, 2016.

[20] Afrodita Iorgulescu. Implicative-groups vs. groups and generalizations. Matrix Rom, Bucharest, 2018.

[21] Afrodita Iorgulescu. Generalizations of MV algebras, ortholattices and Boolean algebras. In The 8th International Workshop on Many-Valued Logic (ManyVal 2019), Bucharest, Romania, 2019.

[22] Afrodita Iorgulescu. Algebras of logic vs. algebras. In Adrian Rezus, editor, Contemporary Logic and Computing, volume 1 of Landscapes in Logic, pages 157-258. College Publications, 2020.

[23] Afrodita Iorgulescu. On quantum-MV algebras – Part II: The orthomodular lattices, semilattices and weaklattices. (submitted).

[24] Afrodita Iorgulescu. On quantum-MV algebras – Part III: The properties (m-Pabs-i) and (WNMm). (submitted).

[25] Afrodita Iorgulescu and Michael 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: Selected Papers from ISMVL 2020, 8(5):1169-1213, 2021.

[26] Afrodita Iorgulescu and Michael Kinyon. Two generalizations of bounded involutive lattices and of ortholattices. Journal of Applied Logics (JALs-ifCoLog), 8(7):2173-2218, 2021.

[27] Afrodita Iorgulescu and Michael Kinyon. Putting quantum MV algebras on the \map” (In memoriam William W. McCune). Scientiae Mathematicae Japonicae, 34:online, 2021. URL: http://www.jams.or.jp/notice/scmjol/2021.html.

[28] Gudrun Kalmbach. Orthomodular Lattices, volume 18 of London Mathematical Society Monographs. Academic Press, New York, 1983.

[29] William W. McCune. Prover9 and Mace4. URL: https://www.cs.unm.edu/~mccune/prover9/.

[30] Carles Noguera, Francesc Esteva, and Joan Gispert. On triangular norm based axiomatic extensions of the Weak Nilpotent Minimum Logic. Mathematical Logic Quarterly, 54(4):387-409, 2008. doi:10.1002/malq.200710054.

[31] Ranganathan Padmanabhan and Sergiu Rudeanu. Axioms for Lattices and Boolean Algebras. World Scientific, Singapore, 2008. doi:10.1142/7007.

Bibtex

@article{sacscuza:iorgulescu21oomaptoa,
  title={On Quantum-MV Algebras - Part I: The Orthomodular Algebras},
  author={A. Iorgulescu},
  journal={Scientific Annals of Computer Science},
  volume={31},
  number={2},
  organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
  year={2021},
  pages={163-222},
  publisher={Alexandru Ioan Cuza University Press, Ia\c si},
  doi={10.7561/SACS.2021.2.163}
}