Algebra universalis

, 79:86 | Cite as

Ordered group-valued probability, positive operators, and integral representations

  • Tomáš KroupaEmail author


Probability maps are additive and normalised maps taking values in the unit interval of a lattice ordered Abelian group. They appear in theory of affine representations and they are also a semantic counterpart of Hájek’s probability logic. In this paper we obtain a correspondence between probability maps and positive operators of certain Riesz spaces, which extends the well-known representation theorem of real-valued MV-algebraic states by positive linear functionals. When the codomain algebra contains all continuous functions, the set of all probability maps is convex, and we prove that its extreme points coincide with homomorphisms. We also show that probability maps can be viewed as a collection of states indexed by maximal ideals of a codomain algebra, and we characterise this collection in special cases.


MV-algebra Abelian \(\ell \)-group Riesz space Positive operator 

Mathematics Subject Classification

06D35 97H50 47H07 



The author is grateful to Prof. Vincenzo Marra (University of Milan) for many suggestions and inspiring comments.


  1. 1.
    Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Springer-Verlag, New York (1971)CrossRefGoogle Scholar
  2. 2.
    Aliprantis, C., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)CrossRefGoogle Scholar
  3. 3.
    Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics, vol. 608. Springer-Verlag, Berlin (1977)CrossRefGoogle Scholar
  4. 4.
    Boccuto, A., Di Nola, A., Vitale, G.: Affine representations of \(\ell \)-groups and MV-algebras. Algebra Univ. 78, 563–577 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning, Trends in Logic-Studia Logica Library, vol. 7. Kluwer Academic Publishers, Dordrecht (2000)zbMATHGoogle Scholar
  6. 6.
    Di Nola, A., Dvurečenskij, A., Jakubík, J.: Good and bad infinitesimals, and states on pseudo MV-algebras. Order 21, 293–314 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Di Nola, A., Leuştean, I.: Łukasiewicz logic and Riesz spaces. Soft Comput. 18, 2349–2363 (2014)CrossRefGoogle Scholar
  8. 8.
    Diestel, J., Uhl, J.J.: Vector Measures. No. 15 in Mathematical Surveys. American Mathematical Society, Providence (1977)CrossRefGoogle Scholar
  9. 9.
    Ellis, A.: Extreme positive operators. Q. J. Math. 15, 342–344 (1964)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Flaminio, T., Kroupa, T.: States of MV-algebras. In: Cintula, P., Fermuller, C., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic, vol. 3. Studies in Logic, Mathematical Logic and Foundations. College Publications, London (2015)zbMATHGoogle Scholar
  11. 11.
    Flaminio, T., Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logics. Internat. J. Approx. Reason 50, 138–152 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, vol. 20. American Mathematical Society, Providence (1986)Google Scholar
  13. 13.
    Hájek, P.: Metamathematics of Fuzzy Logic, Trends in Logic-Studia Logica Library, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  14. 14.
    Holmes, R.B.: Geometric Functional Analysis and Its Applications. Graduate Texts in Mathematics, vol. 24. Springer Verlag, New York (1975)CrossRefGoogle Scholar
  15. 15.
    Kawano, Y.: On the structure of complete MV-algebras. J. Algebra 163, 773–776 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kroupa, T., Marra, V.: Generalised states: a multi-sorted algebraic approach to probability. Soft Comput. 21, 57–67 (2017)CrossRefGoogle Scholar
  17. 17.
    Marra, V., Spada, L.: The dual adjunction between MV-algebras and Tychonoff spaces. Stud. Logica 100, 253–278 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    McNaughton, R.: A theorem about infinite-valued sentential logic. J. Symb. Log. 16, 1–13 (1951)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mundici, D.: Averaging the truth-value in Łukasiewicz logic. Stud. Logica 55, 113–127 (1995)CrossRefGoogle Scholar
  20. 20.
    Mundici, D.: Advanced Łukasiewicz Calculus and MV-algebras, Trends in Logic, p. 35. Springer, Dordrecht (2011)CrossRefGoogle Scholar
  21. 21.
    Panti, G.: Invariant measures in free MV-algebras. Commun. Algebra 36, 2849–2861 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Phelps, R.R.: Extreme positive operators and homomorphisms. Trans. Am. Math. Soc. 108, 265–274 (1963)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Schaefer, H.: Banach Lattices and Positive Operators. Springer-Verlag, New York (1974)CrossRefGoogle Scholar
  24. 24.
    Willard, S.: General Topology. Addison–Wesley Publishing Company, Reading (1970)zbMATHGoogle Scholar
  25. 25.
    Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer Science & Business Media, New York (2012)zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.The Czech Academy of SciencesInstitute of Information Theory and AutomationPragueCzech Republic

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