Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 003, 9 pages      arXiv:1001.1322
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Modularity, Atomicity and States in Archimedean Lattice Effect Algebras

Jan Paseka
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlárská 2, CZ-611 37 Brno, Czech Republic

Received September 29, 2009, in final form January 07, 2010; Published online January 08, 2010

Effect algebras are a generalization of many structures which arise in quantum physics and in mathematical economics. We show that, in every modular Archimedean atomic lattice effect algebra E that is not an orthomodular lattice there exists an (o)-continuous state ω on E, which is subadditive. Moreover, we show properties of finite and compact elements of such lattice effect algebras.

Key words: effect algebra; state; modular lattice; finite element; compact element.

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  1. Avallone A., Barbieri G., Vitolo P., Weber H., Decomposition of effect algebras and the Hammer-Sobczyk theorem, Algebra Universalis 60 (2009), 1-18.
  2. Busch P., Grabowski M., Lahti P.J., Operational quantum physics, Lecture Notes in Physics. New Series m: Monographs, Vol. 31, Springer-Verlag, New York, 1995.
  3. Chajda I., Halas R., Kühr J., Many-valued quantum algebras, Algebra Universalis 60 (2009), 63-90.
  4. Chang C.C., Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467-490.
  5. Chovanec F., Kôpka F., Difference posets in the quantum structures background, Internat. J. Theoret. Phys. 39 (2000), 571-583.
  6. Davey B.A., Priestley H.A., Introduction to lattices and order, 2nd ed., Cambridge University Press, New York, 2002.
  7. Dvurecenskij A., Pulmannová S., New trends in quantum structures, Mathematics and its Applications, Vol. 516, Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000.
  8. Dvurecenskij A., Graziano M.G., An invitation to economical test spaces and effect algebras, Soft Comput. 9 (2005), 463-470.
  9. Epstein L.G., Zhang J., Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001), 265-306.
  10. Foulis D.J., Bennett M.K., Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331-1352.
  11. Foulis D.J., Effects, observables, states, and symmetries in physics, Found. Phys. 37 (2007), 1421-1446.
  12. Greechie R.J., Foulis D.J., Pulmannová S., The center of an effect algebra, Order 12 (1995), 91-106.
  13. Gudder S.P., Sharply dominating effect algebras, Tatra Mt. Math. Publ. 15 (1998), 23-30.
  14. Gudder S.P., S-dominating effect algebras, Internat. J. Theoret. Phys. 37 (1998), 915-923.
  15. Jacobson N., Lectures in abstract algebra, Vol. I, Basic concepts, D. Van Nostrand Co., Inc., Toronto - New York - London, 1951.
  16. Jenca G., Riecanová Z., On sharp elements in lattice ordered effect algebras, BUSEFAL 80 (1999), 24-29.
  17. Kalmbach G., Orthomodular lattices, Mathematics and its Applications, Vol. 453, Kluwer Academic Publishers, Dordrecht, 1998.
  18. Kôpka F., D-posets of fuzzy sets, Tatra Mt. Math. Publ. 1 (1992), 83-87.
  19. Kôpka F., Chovanec F., D-posets, Math. Slovaca 44 (1994), 21-34.
  20. Kôpka F., Compatibility in D-posets, Internat. J. Theoret. Phys. 34 (1995), 1525-1531.
  21. Paseka J., Riecanová Z., Compactly generated de Morgan lattices, basic algebras and effect algebras, Internat. J. Theoret. Phys., to appear.
  22. Paseka J., Riecanová Z., Wu J., Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras, arXiv:0908.3288.
  23. Paseka J., Riecanová Z., The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states, Soft Comput., to appear.
  24. Riecanová Z., Compatibility and central elements in effect algebras, Tatra Mt. Math. Publ. 16 (1999), 151-158.
  25. Riecanová Z., Subalgebras, intervals and central elements of generalized effect algebras, Internat. J. Theoret. Phys. 38 (1999), 3209-3220.
  26. Riecanová Z., Generalization of blocks for D-lattices and lattice-ordered effect algebras, Internat. J. Theoret. Phys. 39 (2000), 231-237.
  27. Riecanová Z., Orthogonal sets in effect algebras, Demonstratio Math. 34 (2001), 525-532.
  28. Riecanová Z., Proper effect algebras admitting no states, Internat. J. Theoret. Phys. 40 (2001), 1683-1691.
  29. Riecanová Z., Lattice effect algebras with (o)-continuous faithful valuations, Fuzzy Sets and Systems 124 (2001), no. 3, 321-327.
  30. Riecanová Z., Smearings of states defined on sharp elements onto effect algebras, Internat. J. Theoret. Phys. 41 (2002), 1511-1524.
  31. Riecanová Z., Continuous lattice effect algebras admitting order-continuous states, Fuzzy Sets and Systems 136 (2003), 41-54.
  32. Riecanová Z., Modular atomic effect algebras and the existence of subadditive states, Kybernetika 40 (2004), 459-468.
  33. Riecanová Z., Paseka J., State smearing theorems and the existence of states on some atomic lattice effect algebras, J. Logic Comput., to appear.
  34. Riecanová Z., Wu J., States on sharply dominating effect algebras, Sci. China Ser. A 51 (2008), 907-914.
  35. Skornyakov L.A., Elements of lattice theory, 2nd ed., Nauka, Moscow, 1982 (English transl.: 1st ed., Hindustan, Delhi; Adam Hilger, Bristol, 1977).
  36. Sykes S.R., Finite modular effect algebras, Adv. in Appl. Math. 19 (1997), 240-250.

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