Pseudo-hoops with Internal States

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


In this chapter we study the internal states for the more general fuzzy structures, namely the pseudo-hoops, and we present state pseudo-hoops and state-morphism pseudo-hoops. We define the notions of state operator, strong state operator, state-morphism operator, weak state-morphism operator and we study their properties. We prove that every strong state pseudo-hoop is a state pseudo-hoop and any state operator on an idempotent pseudo-hoop is a weak state-morphism operator. One of the main results of the chapter consists of proving that every perfect pseudo-hoop admits a state operator. Other results refer to the connection of the state operators with states and generalized states on a pseudo-hoop. Some conditions are given for a state operator to be a generalized state and for a generalized state to be a state operator.


  1. 18.
    Botur, M., Dvurečenskij, A.: State-morphism algebras—general approach. Fuzzy Sets Syst. 218, 90–102 (2013) CrossRefGoogle Scholar
  2. 56.
    Ciungu, L.C.: Algebras on subintervals of pseudo-hoops. Fuzzy Sets Syst. 160, 1099–1113 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 68.
    Ciungu, L.C., Kühr, J.: New probabilistic model for pseudo-BCK algebras and pseudo-hoops. J. Mult.-Valued Log. Soft Comput. 20, 373–400 (2013) Google Scholar
  4. 72.
    Ciungu, L.C., Dvurečenskij, A., Hyčko, M.: State BL-algebras. Soft Comput. 15, 619–634 (2011) zbMATHCrossRefGoogle Scholar
  5. 78.
    Di Nola, A., Dvurečenskij, A.: State-morphism MV-algebras. Ann. Pure Appl. Log. 161, 161–173 (2009) zbMATHCrossRefGoogle Scholar
  6. 79.
    Di Nola, A., Dvurečenskij, A.: On some classes of state-morphism MV-algebras. Math. Slovaca 59, 517–534 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 80.
    Di Nola, A., Dvurečenskij, A.: On varieties of MV-algebras with internal states. Int. J. Approx. Reason. 51, 680–694 (2010) zbMATHCrossRefGoogle Scholar
  8. 101.
    Dvurečenskij, A.: Subdirectly irreducible state-morphism BL-algebras. Arch. Math. Log. 50, 145–160 (2011) zbMATHCrossRefGoogle Scholar
  9. 114.
    Dvurečenskij, A., Rachůnek, J., Šalounová, D.: State operators on generalizations of fuzzy structures. Fuzzy Sets Syst. 187, 58–76 (2012) zbMATHCrossRefGoogle Scholar
  10. 115.
    Dvurečenskij, A., Rachůnek, J., Šalounová, D.: Erratum to “State operators on generalizations of fuzzy structures” [Fuzzy Sets Syst. 187 (2012) 58–76]. Fuzzy Sets Syst. 194, 97–99 (2012) zbMATHCrossRefGoogle Scholar
  11. 119.
    Flaminio, T., Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logics. Int. J. Approx. Reason. 50, 138–152 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 243.
    Rachůnek, J., Šalounová, D.: State operators on GMV-algebras. Soft Comput. 15, 327–334 (2011) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

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

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