Soft Computing

, Volume 22, Issue 12, pp 3879–3889 | Cite as

Ideals and congruences in quasi-pseudo-MV algebras

  • Wenjuan ChenEmail author
  • Wieslaw A. Dudek


Quasi-pseudo-MV algebras (quasi-pMV algebras, for short) were introduced both as the generalization of quasi-MV algebras and as the generalization of pseudo-MV algebras. In the present paper, we mainly investigate ideals and congruences in a quasi-pMV algebra. We present the properties of ideals of quasi-pMV algebras and investigate some special ideals. Furthermore, we study the normal ideals in detail and characterize the bijective correspondence between normal ideals and ideal congruences. Finally, we introduce weak ideals of a quasi-pMV algebra and show the relation existing between normal weak ideals and congruences.


Congruences Ideals Ideal congruences Normal ideals Quasi-pseudo-MV algebras Weak ideals 



This study was funded by the National Natural Science Foundation of China (Grant No.11501245, 11126301) and Natural Scientific Foundation of Shandong Province (No. ZR2013AQ007).

Compliance with ethical standards

Conflict of interest

Authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Bou F, Paoli F, Ledda A, Freytes H (2008) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\)quasi-MV algebras. Part II Soft Comput 12(4):341–352CrossRefzbMATHGoogle Scholar
  2. Bou F, Paoli F, Ledda A, Spinks M, Giuntini R (2010) The logic of quasi-MV algebras. J Logic Comput 20(2):619–643Google Scholar
  3. Chen WJ, Davvaz B (2016) Some classes of quasi-pseudo-MV algebras. Logic J IGPL 24(5):655–673MathSciNetCrossRefGoogle Scholar
  4. Chen WJ, Dudek WA (2015) The representation of square root quasi-pseudo-MV algebras. Soft Comput 19(2):269–282CrossRefzbMATHGoogle Scholar
  5. Chen WJ, Dudek WA (2016) Quantum computational algebra with a non-commutative generalization. Math Slovaca 66(1):19–34MathSciNetzbMATHGoogle Scholar
  6. Dvurecenskij A (2001) On pseudo MV-algebras. Soft Comput 5:347–354CrossRefzbMATHGoogle Scholar
  7. Dvurecenskij A (2002) Pseudo MV-algebras are intervals in \(l\)-groups. J Aust Math Soc 72:427–445MathSciNetCrossRefzbMATHGoogle Scholar
  8. Freytes H, Domenech G (2013) Quantum computational logic with mixed states. Math Logic Q 59(1–2):27–50MathSciNetCrossRefzbMATHGoogle Scholar
  9. Georgescu G, Iorgulescu A (2001) Pseudo MV algebras. MultII Valued Logic 6:95–135MathSciNetzbMATHGoogle Scholar
  10. Giuntini R, Pulmannova S (2000) Ideals and congruences in effect algebras and qmv-algebras. Commun Algebra 28:1567–1592MathSciNetCrossRefzbMATHGoogle Scholar
  11. Jipsen P, Ledda A, Panli F (2013) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part IV Rep Math Logic 48:3–36MathSciNetzbMATHGoogle Scholar
  12. Kowalski T, Paoli F (2010) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part III Rep Math Logic 45:161–199MathSciNetzbMATHGoogle Scholar
  13. Kowalski T, Paoli F (2011) Joins and subdirect products of varieties. Algebra Univers 65:371–391MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kowalski T, Paoli F, Spinks M (2011) Quasi-subtractive varieties. J Symb Logic 76(4):1261–1286MathSciNetCrossRefzbMATHGoogle Scholar
  15. Ledda A, Konig M, Paoli F, Giuntini R (2006) MV algebras and quantum computation. Stud Logica 82:245–270MathSciNetCrossRefzbMATHGoogle Scholar
  16. Paoli F, Ledda A, Giuntini R, Freytes H (2009) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\)quasi-MV algebras. Part I Rep Math Logic 44:31–63MathSciNetzbMATHGoogle Scholar
  17. Pulmannova S, Vincekova E (2009) Congruences and ideals in lattice effect algebras as basic algebras. Kybernetika 45(6):1030–1039MathSciNetzbMATHGoogle Scholar
  18. Rachunek J (2002) A non-commutative generalization of MV-algebras. Czechoslov Math J 52(127):255–273MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of JinanJinanPeople’s Republic of China
  2. 2.Institute of MathematicsWrocław University of TechnologyWrocławPoland

Personalised recommendations