Soft Computing

, Volume 22, Issue 24, pp 8025–8040 | Cite as

States, state operators and quasi-pseudo-MV algebras

  • Wenjuan Chen
  • Wieslaw A. Dudek


Quasi-pseudo-MV algebras (quasi-pMV algebras, for short) arising from quantum computational logics are the generalizations of both quasi-MV algebras and pseudo-MV algebras. In this paper, we introduce the notions of states, state-morphisms, state operators and state-morphism operators to quasi-pMV algebras. First, we present the related properties of states on quasi-pMV algebras and show that states and Bosbach states coincide on any quasi-pMV algebra. And then we investigate the relationship between state-morphisms and the normal and maximal ideals of quasi-pMV algebras. We prove state-morphisms and extremal states are equivalent. The existence of states on quasi-pMV algebras is also discussed. Finally, state operators and state-morphism operators are introduced to quasi-pMV algebras, and the corresponding structures are called state quasi-pMV algebras and state-morphism quasi-pMV algebras, respectively. We investigate the related properties of ideals under state operators and state-morphism operators. Meanwhile, we show that there is a bijective correspondence between normal \(\sigma \)-ideals and ideal congruences on state quasi-pMV algebras.


Ideals Quasi-pseudo-MV algebras States State operators State quasi-pMV algebras 



This study was funded by the National Natural Science Foundation of China (Grant No. 11501245), China Postdoctoral Science Foundation (No. 2017M622177) and Shandong Province Postdoctoral Innovation Projects of Special Funds (No. 201702005).

Compliance with ethical standards

Conflict of interest

Author A declares that she has no conflict of interest. Author B declares that he has no conflict of interest.

Ethical standard

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):341–352CrossRefGoogle Scholar
  2. Bou F, Paoli F, Ledda A, Spinks M, Giuntini R (2010) The logic of quasi-MV algebras. J Logic Comput 20(2):619–643MathSciNetCrossRefGoogle Scholar
  3. Cattaneo G, Dalla Chiara ML, Giuntini R (2004a) An unsharp logic from quantum computation. Int J Theor Phys 43:1803–1817MathSciNetCrossRefGoogle Scholar
  4. Cattaneo G, Dalla Chiara ML, Giuntini R, Leporini R (2004b) Quantum computational structures. Math Slovaca 54:87–108MathSciNetzbMATHGoogle Scholar
  5. Chang CC (1958) Algebraic analysis of many valued logics. Trans Am Math Soc 88:467–490MathSciNetCrossRefGoogle Scholar
  6. Chen WJ, Davvaz B (2016) Some classes of quasi-pseudo-MV algebras. Logic Jnl IGPL 24(5):655–673MathSciNetCrossRefGoogle Scholar
  7. Chen WJ, Dudek WA (2015) The representation of square root quasi-pseudo-MV algebras. Soft Comput 19(2):269–282CrossRefGoogle Scholar
  8. Chen WJ, Dudek WA (2016) Quantum computational algebra with a non-commutative generalization. Math Slovaca 66(1):19–34MathSciNetzbMATHGoogle Scholar
  9. Chen WJ, Dudek WA (2017) Ideals and congruences in quasi-pseudo-MV algebras. Soft Comput. CrossRefGoogle Scholar
  10. Ciungu LC (2008) Bosbach and Riecan states on residuated lattices. J Appl Funct Anal 3(2):175–188MathSciNetzbMATHGoogle Scholar
  11. Ciungu LC, Dvurecenskij A (2009) Measures, states and de Finetti maps on pseudo-BCK algebras. Fuzzy Sets Syst 161:2870–2896MathSciNetCrossRefGoogle Scholar
  12. Ciungu LC (2013) Bounded pseudo-hoops with internal states. Math Slovaca 63:903–934MathSciNetCrossRefGoogle Scholar
  13. Ciungu LC (2015) Internal states on equality algebras. Soft Comput 19(4):939–953CrossRefGoogle Scholar
  14. Di Nola A, Dvurecenskij A (2009) State-morphism MV-algebras. Ann Pure Appl Logic 161:161–173MathSciNetCrossRefGoogle Scholar
  15. Dvurecenskij A (1999) Measures and states on BCK-algebras. Atti del Sem Mat Fisico Univ Modena 47:511–528MathSciNetzbMATHGoogle Scholar
  16. Dvurecenskij A (2001) States on pseudo MV-algebras. Stud Log 68:301–327MathSciNetCrossRefGoogle Scholar
  17. Dvurecenskij A (2002) Pseudo MV-algebras are intervals in \(l\)-groups. J Austral Math Soc 72:427–445MathSciNetCrossRefGoogle Scholar
  18. Dvurecenskij A, Rachunek J (2006a) Probabilistic averaging in bounded commutative residuated \(l\)-monoids. Discrete Math 306:1317–1326MathSciNetCrossRefGoogle Scholar
  19. Dvurecenskij A, Rachunek J (2006b) Probabilistic averaging in bounded non-commutative R\(l\)-monoids. Semigroup Forum 72:190–206MathSciNetCrossRefGoogle Scholar
  20. Dvurecenskij A, Rachunek J (2006c) On Riecan and Bosbach states for bounded non-commutative R\(l\)-monoids. Math Slovaca 56:487–500MathSciNetzbMATHGoogle Scholar
  21. Dymek G, Walendziak A (2007) Semisimple, archimedean, and semilocal pseudo MV-algebras. Scientiae Mathematicae Japonicae Online, pp 315–324Google Scholar
  22. Flaminio T, Montagna F (2009) MV algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50:138–152MathSciNetCrossRefGoogle Scholar
  23. Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras. Mult Val Logic 6:95–135MathSciNetzbMATHGoogle Scholar
  24. Georgescu G (2004) Bosbach states on fuzzy structures. Soft Comput 8:217–230MathSciNetCrossRefGoogle Scholar
  25. Giuntini R, Ledda A, Paoli F (2007) Expanding quasi-MV algebras by a quantum operator. Stud Log 87:99–128MathSciNetCrossRefGoogle Scholar
  26. He PF, Xin XL, Yang YW (2015) On state residuated lattices. Soft Comput 19:2083–2094CrossRefGoogle Scholar
  27. He PF, Zhao B, Xin XL (2017) States and internal states on semihoops. Soft Comput 21(11):2941–2957CrossRefGoogle Scholar
  28. 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
  29. 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
  30. Kowalski T, Paoli F (2011) Joins and subdirect products of varieties. Algebra Univ 65:371–391MathSciNetCrossRefGoogle Scholar
  31. Kowalski T, Paoli F, Spinks M (2011) Quasi-subtractive varieties. J Symb Logic 76(4):1261–1286MathSciNetCrossRefGoogle Scholar
  32. Ledda A, Konig M, Paoli F, Giuntini R (2006) MV algebras and quantum computation. Stud Log 82:245–270MathSciNetCrossRefGoogle Scholar
  33. Leustean I (2001) Local pseudo MV-algebras. Soft Comput 5:386–395CrossRefGoogle Scholar
  34. Mundici D (1995) Averaging the truth-value in Lukasiewicz logic. Stud Log 55:113–127CrossRefGoogle Scholar
  35. 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 44:31–63zbMATHGoogle Scholar
  36. Paoli F, Ledda A, Spinks M, Freytes H, Giuntini G (2011) Logics from \(\sqrt{^{\prime }}\) quasi-MV algebras. Int J Theor Phys 50:3882–3902MathSciNetCrossRefGoogle Scholar
  37. Pulmannova S, Vincekova E (2014) State-morphism pseudo-effect algebras. Soft Comput 18:5–13CrossRefGoogle Scholar
  38. Pulmannova S, Vincekova E (2015) MV-pair and state operators. Fuzzy Set Syst 260:62–76MathSciNetCrossRefGoogle Scholar
  39. Rachunek J (2002) A non-commutative generalization of MV-algebras. Czechoslovak Math J 52(127):255–273MathSciNetCrossRefGoogle Scholar
  40. Rachunek J, Salounova D (2011) State operators on GMV algebras. Soft Comput 15:327–334CrossRefGoogle Scholar
  41. Riecan B (2000) On the probability on BL-algebras. Acta Math Nitra 4:3–13Google Scholar
  42. Turunen E, Mertanen J (2008a) States on semi-divisible residuated lattices. Soft Comput 12:353–357CrossRefGoogle Scholar
  43. Turunen E, Mertanen J (2008b) States on semi-divisible generalized residuated lattices reduce to states on MV-algebras. Fuzzy Sets Syst 159:3051–3064MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

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

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