Soft Computing

, Volume 21, Issue 10, pp 2537–2547 | Cite as

Weak QMV algebras and some ring-like structures

  • Xian Lu
  • Yun ShangEmail author
  • Ru-qian Lu
  • Jian Zhang
  • Feifei Ma


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.


Quantum logic QMV algebras Weak QMV algebras Semirings Bimonoid 



This work has been supported by the National Research Foundation of China (Grant Nos. 61472412, 61621003) and National Key Research and Development Program of China under Grant 2016YFB1000902 and supported in part by the CAS/SAFEA International Partnership Program for Creative Research Teams.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Xian Lu
    • 1
  • Yun Shang
    • 2
    • 3
    Email author
  • 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

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