Measures on Pseudo-BCK Algebras

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


In this chapter we generalize the measures on BCK-algebras introduced by Dvurečenskij and Pulmannova, to pseudo-BCK algebras that are not necessarily bounded. In particular, we show that if A is a downwards-directed pseudo-BCK algebra and m a measure on it, then the quotient over the kernel of m can be embedded into the negative cone of an Abelian, Archimedean -group as its subalgebra. This result will enable us to characterize nonzero measure-morphisms as measures whose kernel is a maximal deductive system.


Borel Measure Weak Topology Deductive System Borel Probability Measure Negative Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 41.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic, Dordrecht (2000) CrossRefzbMATHGoogle Scholar
  2. 76.
    Darnel, M.R.: Theory of Lattice-Ordered Groups. Dekker, New York (1995) zbMATHGoogle Scholar
  3. 94.
    Dvurečenskij, A.: Measures and states on BCK-algebras. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 47, 511–528 (1999) zbMATHGoogle Scholar
  4. 96.
    Dvurečenskij, A.: States on pseudo-MV algebras. Stud. Log. 68, 301–327 (2001) CrossRefzbMATHGoogle Scholar
  5. 103.
    Dvurečenskij, A., Graziano, M.G.: Commutative BCK-algebras and lattice ordered groups. Math. Jpn. 49, 159–174 (1999) zbMATHGoogle Scholar
  6. 108.
    Dvurečenskij, A., Pulmannova, S.: New Trends in Quantum Structures. Kluwer Academic, Dordrecht (2000) CrossRefzbMATHGoogle Scholar
  7. 112.
    Dvurečenskij, A., Vetterlein, T.: Algebras in the positive cone of po-groups. Order 19, 127–146 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 155.
    Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Mathematical Surveys and Monographs, vol. 20. Am. Math. Soc., Providence (1986) zbMATHGoogle Scholar
  9. 202.
    Kroupa, T.: Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst. 157, 2771–2782 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 208.
    Kühr, J.: Pseudo-BCK algebras and related structures. Univerzita Palackého v Olomouci (2007) Google Scholar
  11. 211.
    Kühr, J., Mundici, D.: De Finetti theorem and Borel states in [0,1]-valued algebraic logic. Int. J. Approx. Reason. 46, 605–616 (2007) CrossRefzbMATHGoogle 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