Advertisement

States on EMV-algebras

  • Anatolij DvurečenskijEmail author
  • Omid Zahiri
Foundations
  • 10 Downloads

Abstract

We define a state as a [0, 1]-valued, finitely additive function attaining the value 1 on an EMV-algebra, which is an algebraic structure close to MV-algebras, where the top element is not assumed. The state space of an EMV-algebra is a convex space that is not necessarily compact, and in such a case, the Krein–Mil’man theorem cannot be used. Nevertheless, we show that the set of extremal states generates the state space. We show that states always exist and the extremal states are exactly state-morphisms. Nevertheless, the state space is a convex space that is not necessarily compact; a variant of the Krein–Mil’man theorem, saying states are generated by extremal states, is proved. We define a weaker form of states, pre-states and strong pre-states, and also Jordan signed measures which form a Dedekind complete \(\ell \)-group. Finally, we show that every state can be represented by a unique regular Borel probability measure, and a variant of the Horn–Tarski theorem is proved.

Keywords

MV-algebra EMV-algebra State State-morphism Krein–Mil’man representation Pre-state Strong pre-state Jordan signed measure Integral representation of states The Horn–Tarski theorem 

Notes

Funding

Author A.D. has received research Grants from the Slovak Research and Development Agency under contract APVV-16-0073 and the Grant VEGA No. 2/0069/16 SAV.

Compliance with ethical standards

Conflicts of interest

All authors have declared that they have no conflict of interest.

Animals and human participants

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Alfsen EM (1971) Compact convex sets and boundary integrals. Springer, BerlinCrossRefzbMATHGoogle Scholar
  2. Chang CC (1958) Algebraic analysis of many valued logics. Trans Am Math Soc 88:467–490MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer Academic Publ, DordrechtCrossRefzbMATHGoogle Scholar
  4. Conrad P, Darnel MR (1998) Generalized Boolean algebras in lattice-ordered groups. Order 14:295–319MathSciNetCrossRefzbMATHGoogle Scholar
  5. de Finetti B (1993) Sul significato soggettivo della probabilitá. Fundam Math 17 (1931), 298–329. Translated into English as On the subjective meaning of probability (Monari P, Cocchi D (eds), Trans.). Probabilitá e Induzione. Clueb, Bologna, pp 291–321Google Scholar
  6. de Finetti B (1949) Sull impostazione assiomatica del calcolo delle probabilità. Annali Tiestini Sez II(19):19–81Google Scholar
  7. de Finetti B (1950) Aggiunta alla nota sull’assiomatica del calcolo delle probabilità. Annali Tiestini Sez II(20):5–22Google Scholar
  8. de Finetti B (1974) Theory of probability, vol 1. Wiley, ChichesterzbMATHGoogle Scholar
  9. Dubins LE, Savage LJ (1965) How to gamble if you must: inequalities for stochastic processes. McGraw-Hill, LondonzbMATHGoogle Scholar
  10. Dvurečenskij A (2001) States on pseudo MV-algebras. Stud Log 68:301–327MathSciNetCrossRefzbMATHGoogle Scholar
  11. Dvurečenskij A (2003) Central elements and Cantor-Bernstein’s theorem for pseudo-effect algebras. J Austral Math Soc 74:121–143MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dvurečenskij A (2010) Every state on interval effect algebra is integral. J Math Phys 51:083508–12.  https://doi.org/10.1063/1.3467463 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dvurečenskij A (2011) States on quantum structures versus integrals. Int J Theor Phys 50:3761–3777MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dvurečenskij A, Zahiri O (2018) Morphisms on EMV-algebras and their applications. Soft Comput 22:7519–7537.  https://doi.org/10.1007/s00500-018-3039-7 CrossRefzbMATHGoogle Scholar
  15. Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Academic Publ., Dordrecht pp 541 + xviGoogle Scholar
  16. Dvurečenskij A, Zahiri O, Loomis–Sikorski theorem for \(\sigma \)-complete EMV-algebras. J Australian Math Soc.  https://doi.org/10.1017/S1446788718000101
  17. Dvurečenskij A, Zahiri O, On EMV-algebras, arXiv:1706.00571
  18. Flaminio T, Montagna F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50:138–152MathSciNetCrossRefzbMATHGoogle Scholar
  19. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346MathSciNetCrossRefzbMATHGoogle Scholar
  20. Georgescu G (2004) Bosbach states on fuzzy structures. Soft Comput 8:217–230MathSciNetCrossRefzbMATHGoogle Scholar
  21. Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras. Multi Val Log 6:95–135MathSciNetzbMATHGoogle Scholar
  22. Goodearl KR (1986) Partially ordered abelian groups with interpolation. Math Surv Monogr (No. 20). American Mathematical Soc., Providence, Rhode IslandGoogle Scholar
  23. Halmos PR (1988) Measure theory. Springer, BerlinGoogle Scholar
  24. Horn A, Tarski A (1948) Measures on Boolean algebras. Trans Am Math Soc 64:467–497MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kelley JL (1955) General topology. Van Nostrand, PrincetonzbMATHGoogle Scholar
  26. Kolmogorov AN (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, BerlinCrossRefzbMATHGoogle Scholar
  27. Kroupa T (2006) Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst 157:2771–2782MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kroupa T (2006) Representation and extension of states on MV-algebras. Arch Math Logic 45:381–392MathSciNetCrossRefzbMATHGoogle Scholar
  29. Mundici D (1986) Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus. J Func Anal 65:15–63CrossRefzbMATHGoogle Scholar
  30. Mundici D (1995) Averaging the truth-value in Łukasiewicz logic. Stud Log 55:113–127CrossRefzbMATHGoogle Scholar
  31. Mundici D (2009) Interpretation of de Finetti coherence criterion in Łukasiewicz Logic. Ann Pure Appl Log 161:235–245CrossRefzbMATHGoogle Scholar
  32. Panti G (2008) Invariant measures in free MV-algebras. Commun Algebra 36:2849–2861MathSciNetCrossRefzbMATHGoogle Scholar
  33. Rao KB, Rao MB (1983) Theory of charges: a study of finitely additive measures. Academic Press, LondonzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical Institute, Slovak Academy of SciencesBratislavaSlovakia
  2. 2.Faculty of Sciences, Palacký University OlomoucOlomoucCzech Republic
  3. 3.University of Applied Science and TechnologyTehranIran

Personalised recommendations