, Volume 27, Issue 1, pp 41–61 | Cite as

Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras



We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element below x by x . For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x ,x] is a subset of B. For every meager element (that means, an element x with x  = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2M(E) given by h(a) = [0,a] ∩ M(E).


Effect algebra Orthomodular lattice BCK-algebra 

Mathematics Subject Classifications (2000)

Primary 06C15; Secondary 03G12 81P10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bennett, M.K., Foulis, D.J.: Interval and scale effect algebras. Adv. Appl. Math. 19, 200–215 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cattaneo, G.: A unified framework for the algebra of unsharp quantum mechanics. Int. J. Theor. Phys. 36, 3085–3117 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1959)CrossRefGoogle Scholar
  4. 4.
    Chen, C.C., Grätzer, G.: Stone lattices I. Construction theorems. Can. J. Math. 21, 884–894 (1969)MATHGoogle Scholar
  5. 5.
    Chen, C.C., Grätzer, G.: Stone lattices II. Structure theorems. Can. J. Math. 21, 895–903 (1969)MATHGoogle Scholar
  6. 6.
    Chevalier, G., Pulmannová, S.: Some ideal lattices in partial abelian monoids. Order 17, 75–92 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chovanec, F., Kôpka, F.: Boolean D-posets. Tatra Mt. Math. Publ. 10, 183–197 (1997)MATHMathSciNetGoogle Scholar
  8. 8.
    Dvurečenskij, A.: Effect algebras which can be covered by MV-algebras. Int. J. Theor. Phys. 41, 221–229 (2002)MATHCrossRefGoogle Scholar
  9. 9.
    Dvurečenskij, A., Graziano, M.G.: Remarks on representations of minimal clans. Tatra Mt. Math. Publ. 15, 31–53 (1998)MATHMathSciNetGoogle Scholar
  10. 10.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer and Ister Science, Dordrecht (2000)MATHGoogle Scholar
  11. 11.
    Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Foulis, D.J., Greechie, R., Rütimann, G.: Filters and supports in orthoalgebras. Int. J. Theor. Phys. 35, 789–802 (1995)Google Scholar
  13. 13.
    Giuntini, R., Greuling, H.: Toward a formal language for unsharp properties. Found. Phys. 19, 931–945 (1989)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Greechie, R., Foulis, D., Pulmannová, S.: The center of an effect algebra. Order 12, 91–106 (1995)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gudder, S.: Sharply dominating effect algebras. Tatra Mt. Math. Publ. 15, 23–30 (1998)MATHMathSciNetGoogle Scholar
  16. 16.
    Jenča, G.: Subcentral ideals in generalized effect algebras. Int. J. Theor. Phys. 39, 745–755 (2000)MATHCrossRefGoogle Scholar
  17. 17.
    Jenča, G.: Blocks of homogeneous effect algebras. Bull. Aust. Math. Soc. 64, 81–98 (2001)MATHCrossRefGoogle Scholar
  18. 18.
    Jenča, G.: Finite homogeneous and lattice ordered effect algebras. Discrete Math. 272, 197–214 (2003)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jenča, G., Pulmannová, S.: Quotients of partial abelian monoids and the Riesz decomposition property. Algebra Univers. 47, 443–477 (2002)MATHCrossRefGoogle Scholar
  20. 20.
    Jenča, G., Pulmannová, S.: Orthocomplete effect algebras. Proc. Am. Math. Soc. 131, 2663–2671 (2003)MATHCrossRefGoogle Scholar
  21. 21.
    Jenča, G., Riečanová, Z.: On sharp elements in lattice ordered effect algebras. BUSEFAL 80, 24–29 (1999)Google Scholar
  22. 22.
    Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994)MATHMathSciNetGoogle Scholar
  23. 23.
    Mundici, D.: Interpretation of AF C *-algebras in Lukasziewicz sentential calculus. J. Funct. Anal. 65, 15–53 (1986)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Mundici, D.: MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Math. Jpn. 6, 889–894 (1986)MathSciNetGoogle Scholar
  25. 25.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991)MATHGoogle Scholar
  26. 26.
    Riečanová, Z.: A generalization of blocks for d-lattices and lattice effect algebras. Int. J. Theor. Phys. 39, 231–237 (2000)MATHCrossRefGoogle Scholar
  27. 27.
    Riečanová, Z.: Continuous lattice effect algebras admitting order continuous states. Fuzzy Sets Syst. 136, 41–54 (2003)MATHCrossRefGoogle Scholar
  28. 28.
    Yutani, H.: The class of commutative BCK-algebras is equationally definable. Math. Semin. Notes 5, 207–210 (1977)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Descriptive GeometryFaculty of Civil EngineeringBratislavaSlovak Republic

Personalised recommendations