States on Multiple-Valued Logic Algebras

  • Lavinia Corina Ciungu
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter we will present the notion of state for the case of pseudo-BCK algebras. One of the main results consists of proving that any Bosbach state on a good pseudo-BCK algebra is a Riečan state, but conversely it turns out not to be true. Some conditions are given for a Riečan state on a good pseudo-BCK algebra to be a Bosbach state. In contrast to the case of pseudo-BL algebras, we show that there exist linearly ordered pseudo-BCK algebras having no Bosbach states and that there exist pseudo-BCK algebras having normal deductive systems which are maximal, but having no Bosbach states. Some specific properties of states on FL w -algebras, pseudo-MTL algebras, bounded R-monoids and subinterval algebras of pseudo-hoops are proved. A special section is dedicated to the existence of states on the residuated structures, showing that every perfect FL w -algebra admits at least a Bosbach state and every perfect pseudo-BL algebra has a unique state-morphism. Finally, we introduce the notion of a local state on a perfect pseudo-MTL algebra and we prove that every local state can be extended to a Riečan state.


  1. 17.
    Bosbach, B.: Residuation groupoids. Results Math. 5, 107–122 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 38.
    Chang, C.C.: Algebraic analysis of many-valued logic. Trans. Am. Math. Soc. 88, 467–490 (1958) zbMATHCrossRefGoogle Scholar
  3. 41.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic, Dordrecht (2000) zbMATHCrossRefGoogle Scholar
  4. 49.
    Ciungu, L.C.: Bosbach and Riečan states on residuated lattices. J. Appl. Funct. Anal. 2, 175–188 (2008) MathSciNetGoogle Scholar
  5. 52.
    Ciungu, L.C.: States on pseudo-BCK algebras. Math. Rep. (Bucur.) 10, 17–36 (2008) MathSciNetGoogle Scholar
  6. 67.
    Ciungu, L.C., Dvurečenskij, A.: Measures, states and de Finetti maps on pseudo-BCK algebras. Fuzzy Sets Syst. 160, 1099–1113 (2010) CrossRefGoogle Scholar
  7. 89.
    Di Nola, A., Georgescu, G., Leuştean, I.: MV-algebras. Manuscript Google Scholar
  8. 96.
    Dvurečenskij, A.: States on pseudo-MV algebras. Stud. Log. 68, 301–327 (2001) zbMATHCrossRefGoogle Scholar
  9. 98.
    Dvurečenskij, A.: Every linear pseudo-BL algebra admits a state. Soft Comput. 6, 495–501 (2007) CrossRefGoogle Scholar
  10. 104.
    Dvurečenskij, A., Hyčko, M.: On the existence of states for linear pseudo-BL algebras. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 53, 93–110 (2005) MathSciNetzbMATHGoogle Scholar
  11. 108.
    Dvurečenskij, A., Pulmannova, S.: New Trends in Quantum Structures. Kluwer Academic, Dordrecht (2000) zbMATHCrossRefGoogle Scholar
  12. 109.
    Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded commutative R-monoids. Discrete Math. 306, 1317–1326 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 110.
    Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded non-commutative R-monoids. Semigroup Forum 72, 190–206 (2006) MathSciNetGoogle Scholar
  14. 111.
    Dvurečenskij, A., Rachůnek, J.: On Riečan and Bosbach states for bounded non-commutative R-monoids. Math. Slovaca 56, 487–500 (2006) MathSciNetzbMATHGoogle Scholar
  15. 131.
    Georgescu, G.: Bosbach states on fuzzy structures. Soft Comput. 8, 217–230 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 155.
    Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Mathematical Surveys and Monographs, vol. 20. Am. Math. Soc., Providence (1986) zbMATHGoogle Scholar
  17. 203.
    Kroupa, T.: Representation and extension of states on MV-algebras. Arch. Math. Log. 45, 381–392 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 208.
    Kühr, J.: Pseudo-BCK algebras and related structures. Univerzita Palackého v Olomouci (2007) Google Scholar
  19. 230.
    Mundici, D.: Averaging the truth-value in Łukasiewicz logic. Stud. Log. 55, 113–127 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 245.
    Riečan, B.: On the probability on BL-algebras. Acta Math. Nitra 4, 3–13 (2000) Google Scholar
  21. 249.
    Riečan, B., Mundici, D.: Probability on MV-algebras. In: Pap, E. (ed.) Handbook of Measure Theory, pp. 869–909. North-Holland, Amsterdam (2002) Google Scholar
  22. 250.
    Riečan, B., Neubrunn, T.: Integral, Measure and Ordering. Kluwer Academic, Dordrecht (1997) zbMATHGoogle Scholar
  23. 260.
    Turunen, E., Sessa, S.: Local BL-algebras. Mult. Valued Log. 6, 229–249 (2001) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Lavinia Corina Ciungu
    • 1
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA

Personalised recommendations