Skip Navigation



Journal of Logic and Computation Advance Access published online on March 18, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp011
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Dvurecenskij, A.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

On States on MV-algebras and their Applications

Anatolij Dvurecenskij

Mathematical Institute, Slovak Academy of Sciences, Stefánikova 49, SK-814 73 Bratislava, Slovakia.
E-mail: dvurecen{at}mat.savba.sk

Received 4 September 2008.

We present the notion of a state, as averaging or a probabilistic assessment in many-valued reasoning. We show what a state can be in different algebraic structures, and also present a new trends using de Finetti's coherence principle or state MV-algebras where the state is an internal notion.

Keywords: MV-algebra; unital {ell}-group; representation; state; state-morphism; coherence; state MV-algebra; state-orphism MV-algebra



References

  1. Aarnes JF. Physical states on a C*–algebra. Acta Mathematica (1969) 122:161–172.[CrossRef]
  2. Alfsen EM. Compact Convex Sets and Boundary Integrals (1971) Springer.
  3. Alfsen EM, Schultz FW. State Spaces of Operator Algebras (2001) Birhauser.
  4. Belluce LP. Semisimple algebras of infinite valued logic and bold fuzzy set theory. Canadian Journal of Mathematics (1986) 38:1356–1379.
  5. Birkhoff G, von Neumann J. The logic of quantum mechanics. Annals of Mathematics (1936) 37:823–834.[CrossRef][Web of Science]
  6. Boole G. An Investigation of the Laws of Thought (1854) Macmillan. (reprinted by Dover Press, 1967).
  7. Buhagiar D, Chetcuti E, Dvurecenskij A. Loomis-Sikorski representation of monotone {sigma}-complete effect algebras. Fuzzy Sets and Systems (2006) 157:683–690.[CrossRef][Web of Science]
  8. Buhagiar D, Chetcuti E, Dvurecenskij A. Algebraic and measure-theoretic properties of classes of subspaces of an inner product space. In: Handbook of Quantum Logic and Quantum Structures, Quantum Structures—Engesser K, Gabbay Dov M, Lehmann D, eds. (2007) Elsevier. 75–120.
  9. Bunce LJ, Saitô K, Wright JDM. A Dixmier-Schaefer-Zhang theorem for operator algebras. Proceedings of the American Mathematical Society (1999) 127:2975–2979.[CrossRef][Web of Science]
  10. Chang CC. Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society (1958) 88:467–490.[CrossRef]
  11. Cignoli R, D’Ottaviano IML, Mundici D. Algebraic Foundations of Many-valued Reasoning (2000) Kluwer Academic Publ.
  12. Darnel MR. Theory of Lattice-Ordered Groups (1995) Inc.: Marcel Dekker.
  13. Di Nola A, Dvurecenskij A. State-morphism MV-algebras. In: Journal of Pure and Applied Logic. (in press).
  14. Di Nola A, Dvurecenskij A. On some classes of state-morphismMV-algebras. In: Mathematica Slovaca. (in press).
  15. Di Nola A, Dvurecenskij A, Tsinakis C. On perfect GMV-algebras. Communications in Algebra (2008) 36:1221–1249.[CrossRef][Web of Science]
  16. Di Nola A, Georgescu G, Iorgulescu A. Pseudo-BL algebras I. Multiple Valued Logic (2002) 8:673–714.
  17. Di Nola A, Georgescu G, Iorgulescu A. Pseudo-BL algebras II. Multiple Valued Logic (2002) 8:715–750.
  18. Dvurecenskij A. Gleason's Theorem and Its Applications (1993) Kluwer Academic Publ. and Ister Science Press.
  19. Dvurecenskij A. Loomis–Sikorski theorem for {sigma}-complete MV-algebras and {ell}-groups. Journal of the Autsralian Mathematical Society Series A (2000) 68:261–277.
  20. Dvurecenskij A. States on pseudo MV-algebras. Studia Logica (2001) 68:301–327.[CrossRef]
  21. Dvurecenskij A. Pseudo MV-algebras are intervals in {ell}-groups. Journal of the Autsralian Mathematical Society Series (2002) 72:427–445.
  22. Dvurecenskij A. Every linear pseudo BL-algebra admits a state. Soft Computing (2007) 11:495–501.[CrossRef][Web of Science]
  23. Dvurecenskij A, Holland WC. Top varieties of generalizedMV-algebras and unital latticeordered groups. Communications in Algebra (2007) 35:3370–3390.[CrossRef][Web of Science]
  24. Dvurecenskij A, Pulmannová S. New Trends in Quantum Structures (2000) Kluwer Acad. Publ. and Ister Science.
  25. Dvurecenskij A, Rachunek J. On Riecan and Bosbach states for bounded non-commutative R{ell}-monoids. Mathematica Slovaca (2006) 56:487–500.
  26. Dvurecenskij A, Vetterlein T. Pseudo-effect algebras. I. Basic properties. International Journal of Theoretical Physics (2001) 40:685–701.[CrossRef][Web of Science]
  27. Dvurecenskij A, Vetterlein T. Pseudo-effect algebras. II. Group representation. International Journal of Theoretical Physics (2001) 40:703–726.[CrossRef][Web of Science]
  28. Flaminio T, Montagna F. An algebraic approach to states onMV-algebras. Novák V, ed. II:201–206. In Fuzzy Logic 2, Proceedings of the 5th EUSFLAT Conference, September 11–14, 2007.
  29. Flaminio T, Montagna F. MV-algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning (2009) 50:138–152.[CrossRef][Web of Science]
  30. Foulis DJ, Bennett MK. Effect algebras and unsharp quantum logics. Foundations of Physics (1994) 24:1325–1346.
  31. Georgescu G. Bosbach states on fuzzy structures. Soft Computing (2004) 8:217–230.[Web of Science]
  32. Georgescu G, Iorgulescu A. Pseudo-MV algebras. Multiple Valued Logic (2001) 6:95–135.
  33. Giuntini R, Greuling H. Toward a formal language for unsharp properties. Foundations of Physics (1989) 19:931–945.[CrossRef][Web of Science]
  34. Gleason AM. Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics (1957) 6:885–893.[Web of Science]
  35. Goodearl KR. Partially Ordered Abelian Groups with Interpolation. In: Math. Surveys and Monographs No. 20 (1986) Amerian Mathematical Society.
  36. Greechie R. Orthomodular lattices admitting no states. Journal of Combinatorial Theory (1971) 10:119–132.[CrossRef]
  37. Hamhalter J, Pták P. A completeness criterion for inner product spaces. Bulletin of the London Mathematical Society (1987) 19:259–263.[Abstract/Free Full Text]
  38. Heisenberg W. The Physical Principles of Quantum Theory (1930) New York: Dover.
  39. Kolmogorov AN. Grundbegriffe der Wahrscheinlichkeitsrechnung (1933) Berlin.
  40. Kôpka F, Chovanec F. D-posets. Mathematica Slovaca (1994) 44:21–34.
  41. Kroupa T. Every state on semisimple MV-algebra is integral. Fuzzy Sets and Systems (2006) 157:2771–2782.[CrossRef][Web of Science]
  42. Kühr J, Mundici D. De Finetti theorem and Borel states in [0,1]-valued algebraic logic. International Journal Approximate Reasoning (2007) 46:605–616.[CrossRef]
  43. Mundici D. Interpretation of AF C*-algebras in Lukasiewicz sentential calculus. Journal of Functional Analysis (1986) 65:15–63.[CrossRef][Web of Science]
  44. Mundici D. Averaging the truth-value in Lukasiewicz logic. Studia Logica (1995) 55:113–127.[CrossRef]
  45. Mundici D. Tensor products and the Loomis–Sikorski theorem for MV-algebras. Advances in Applied Mathematics (1999) 22:227–248.[CrossRef][Web of Science]
  46. Panti G. Invariant measures in free MV-algebras. In: Communications in Algebra. (in press).
  47. Phelps RR. Lectures on Choquet's Theorem (1965) Princeton: Van Nostrand.
  48. Rachunek J. A non-commutative generalization of MV-algebras. Czechoslovak Mathematical Journal (2002) 52:255–273.[CrossRef][Web of Science]
  49. Ravindran K. On a structure theory of effect algebras (1996) Kansas State University. PhD Theses.
  50. Riecan B. On the probability on BL-algebras. Acta Mathematica Nitra (2000) 4:3–13.
  51. Sikorski R. Boolean Algebras (1964) Berlin, Heidelberg, New York: Springer.
  52. von Neumann J. Mathematische Grundlagen der Quantenmechanik (1932) Springer.

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Dvurecenskij, A.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?