Soft Computing

, Volume 21, Issue 10, pp 2537–2547

Weak QMV algebras and some ring-like structures

  • Xian Lu
  • Yun Shang
  • Ru-qian Lu
  • Jian Zhang
  • Feifei Ma
Focus

Abstract

In this work, we propose a new quantum structure—weak quantum MV algebras (wQMV algebras)—and define coupled bimonoids and strong coupled bimonoids. We find that the coupled bimonoids and strong coupled bimonoids are ring-like structures corresponding to lattice-ordered wQMV algebras and lattice-ordered QMV algebras, respectively. Using an automated reasoning tool, we give the smallest 4-element wQMV algebra but not a QMV algebra. We also show that lattice-ordered wQMV algebras are the real nondistributive generalization of MV algebras. Certainly, most important properties of quantum MV algebras (QMV algebras) are preserved by wQMV algebras. Furthermore, we can conclude that lattice-ordered wQMV algebras are the simplest unsharp quantum logical structures by far, based on which computation theory could be set up.

Keywords

Quantum logic QMV algebras Weak QMV algebras Semirings Bimonoid 

References

  1. Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37(4):823–843MathSciNetCrossRefMATHGoogle Scholar
  2. Chang CC (1958) Algebraic analysis of many valued logics. Trans Am Math Soc 88:467–490MathSciNetCrossRefMATHGoogle Scholar
  3. Chang CC (1959) A new proof of the completeness of the Łukasiewicz axioms. Trans Am Math Soc 93:74–80MATHGoogle Scholar
  4. Chiara MD, Giuntini R, Greechie R (2004) Reasoning in quantum theory—sharp and unsharp quantum logic. Kluwer Academic Publishers, DordrechtMATHGoogle Scholar
  5. Di Nola A, Gerla B (2005) Algebras of łukasiewicz logic and their semiring reducts. In: Litvinov GL, Maslov VP (eds) Idempotent mathematics and mathematical physics. Contemp. Math., vol 377, pp 131–144. American Mathematical Society, ProvidenceGoogle Scholar
  6. Dvurečenskig A, Pulmannová S (2000) New trends in quantum structures. Kluwer/Ister Science, Dordrecht/BratislavaCrossRefGoogle Scholar
  7. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24(10):1331–1352MathSciNetCrossRefMATHGoogle Scholar
  8. Gerla B (2003) Many-valued logic and semirings. Neural Netw World 5:467–480Google Scholar
  9. Giuntini R (1996) Quantum MV algebras. Studia Logica 56:393–417MathSciNetCrossRefMATHGoogle Scholar
  10. Giuntini R (1998) Quantum MV-algebras and commutativity. Int J Theor Phys 37(1):65–74MathSciNetCrossRefMATHGoogle Scholar
  11. Giuntini R (2005) Weakly linear quantum MV-algebras. Algebra Univers 53:45–72MathSciNetCrossRefMATHGoogle Scholar
  12. Giuntini R, Greuling H (1989) Toward a formal language for unsharp properties. Found Phys 20:931–935MathSciNetCrossRefGoogle Scholar
  13. Gudder S (1995) Total extensions of effect algebras. Found Phys Lett 8:243–252MathSciNetCrossRefGoogle Scholar
  14. Kalmbach G (1983) Orthomodular lattices. Academic Press, New YorkMATHGoogle Scholar
  15. Kôpka F, Chovanec F (1994) D-posets. Math Slovaca 44:21–34MathSciNetMATHGoogle Scholar
  16. Ludwig G (1983) Foundations of quantum mechanics, vol 1. Springer, BerlinMATHGoogle Scholar
  17. Qiu DW (2004) Automata theory based on quantum logic: some characterizations. Inf Comput 190(2):179–195MathSciNetCrossRefMATHGoogle Scholar
  18. Qiu DW (2007) Automata theory based on quantum logic: reversibilities and pushdown automata. Theor Comput Sci 386:38–56MathSciNetCrossRefMATHGoogle Scholar
  19. Shang Y, Lu RQ (2007) Semirings and pseudo MV algebras. Soft Comput 11:847–853CrossRefMATHGoogle Scholar
  20. Shang Y, Lu X, Lu RQ (2009) Automata theory based on unsharp quantum logic. Math Struct Comput Sci 19:737–756MathSciNetCrossRefMATHGoogle Scholar
  21. Shang Y, Lu X, Lu RQ (2012) A theory of computation based on unsharp quantum logic: finite state automata and pushdown automata. Theor Comput Sci 434:53–86MathSciNetCrossRefMATHGoogle Scholar
  22. Ying MS (2000a) Automata theory based on quantum logic (I). Int J Theor Phys 39(4):981–991MathSciNetCrossRefGoogle Scholar
  23. Ying MS (2000b) Automata theory based on quantum logic (II). Int J Theor Phys 39(11):2545–2557MathSciNetCrossRefMATHGoogle Scholar
  24. Ying MS (2005) A theory of computation based on quantum logic (I). Theor Comput Sci 344(2,3):134–207MathSciNetCrossRefMATHGoogle Scholar
  25. Zhang J, Zhang H (1995) SEM: a system for enumerating models. In: Proceedings of international joint conference on AI (IJCAI-95), pp 298–303Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Xian Lu
    • 1
  • Yun Shang
    • 2
    • 3
  • Ru-qian Lu
    • 2
    • 3
  • Jian Zhang
    • 1
    • 4
  • Feifei Ma
    • 1
    • 4
  1. 1.Institute of SoftwareCASBeijingPeople’s Republic of China
  2. 2.Institute of Mathematics, AMSSCASBeijingPeople’s Republic of China
  3. 3.NCMIS, AMSSCASBeijingPeople’s Republic of China
  4. 4.University of CAS, CASBeijingPeople’s Republic of China

Personalised recommendations