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


SIGMA 6 (2010), 001, 8 pages      arXiv:1001.0950      http://dx.doi.org/10.3842/SIGMA.2010.001
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

Zdenka Riecanová
Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovicova 3, SK-812 19 Bratislava, Slovak Republic

Received September 29, 2009, in final form January 04, 2010; Published online January 06, 2010

Abstract
We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra E is separable and modular then there exists a faithful state on E. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra ^E and the compatiblity center of E is not a Boolean algebra then there exists an (o)-continuous subadditive state on E.

Key words: effect algebra; state; sharp element; center; compatibility center.

pdf (207 kb)   ps (147 kb)   tex (14 kb)

References

  1. Boole G., An investigation of the laws of thought, Macmillan, Cambridge, 1854 (reprinted by Dover Press, New York, 1967).
  2. Busch P., Lahti P.J., Mittelstaedt P., The quantum theory of measurement, Lecture Notes in Physics, New Series m: Monographs, Vol. 2, Springer-Verlag, Berlin, 1991.
  3. Dvurecenskij A., Pulmannová S., New trends in quantum structures, Mathematics and its Applications, Vol. 516, Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000.
  4. Foulis D.J., Bennett M.K., Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331-1352.
  5. Foulis D.J., Effects, observables, states, and symmetries in physics, Found. Phys. 37 (2007), 1421-1446.
  6. Greechie R.J., Orthomodular lattices admitting no states, J. Combinatorial Theory Ser. A 10 (1971), 119-132.
  7. Greechie R.J., Foulis D.J., Pulmannová S., The center of an effect algebra, Order 12 (1995), 91-106.
  8. Gudder S.P., Sharply dominating effect algebras, Tatra Mt. Math. Publ. 15 (1998), 23-30.
  9. Gudder S.P., S-dominating effect algebras, Internat. J. Theoret. Phys. 37 (1998), 915-923.
  10. Jenca G., Pulmannová S., Orthocomplete effect algebras, Proc. Amer. Math. Soc. 131 (2003), 2663-2671.
  11. Jenca G., Riecanová Z., On sharp elements in lattice ordered effect algebras, BUSEFAL 80 (1999), 24-29.
  12. Kalina M., On central atoms of Archimedean atomic lattice effect algebras, submitted.
  13. Kôpka F., Chovanec F., Boolean D-posets, Internat. J. Theoret. Phys. 34 (1995), 1297-1302.
  14. Ludwig G., Die Grundlagen der Quantenmechanik, Springer-Verlag, Berlin, 1954 (translated by C.A. Hein, Springer-Verlag, Berlin, 1983).
  15. Mosná K., Atomic lattice effect algebras and their sub-lattice effect algebras, J. Electrical Engineering 58 (2007), 7/S, 3-6.
  16. Paseka J., Riecanová Z., The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states, Soft Comput., to appear.
  17. Riecanová Z., Subalgebras, intervals and central elements of generalized effect algebras, Internat. J. Theoret. Phys. 38 (1999), 3209-3220.
  18. Riecanová Z., MacNeille completions of D-posets and effect algebras, Internat. J. Theoret. Phys. 39 (2000), 859-869.
  19. Riecanová Z., Archimedean and block-finite lattice effect algebras, Demonstratio Math. 33 (2000), 443-452.
  20. Riecanová Z., Generalization of blocks for D-lattices and lattice-ordered effect algebras, Internat. J. Theoret. Phys. 39 (2000), 231-237.
  21. Riecanová Z., Orthogonal sets in effect algebras, Demonstratio Math. 34 (2001), 525-532.
  22. Riecanová Z., Proper effect algebras admitting no states, Internat. J. Theoret. Phys. 40 (2001), 1683-1691.
  23. Riecanová Z., Lattice effect algebras with (o)-continuous faithful valuations, Fuzzy Sets and Systems 124 (2001), no. 3, 321-327.
  24. Riecanová Z., Smearings of states defined on sharp elements onto effect algebras, Internat. J. Theoret. Phys. 41 (2002), 1511-1524.
  25. Riecanová Z., Subdirect decompositions of lattice effect algebras, Internat. J. Theoret. Phys. 42 (2003), 1415-1433.
  26. Riecanová Z., Wu J., States on sharply dominating effect algebras, Sci. China Ser. A 51 (2008), 907-914.
  27. Sarymsakov T.A., Ayupov Sh.A., Khadzhiev Dzh., Chilin V.I., Ordered algebras, FAN, Tashkent, 1983 (in Russian).
  28. Schmidt J., Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Hülle, Arch. Math. (Basel) 7 (1956), 241-249.

Next article   Contents of Volume 6 (2010)