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Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras

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Abstract

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).

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References

  1. Bennett, M.K., Foulis, D.J.: Interval and scale effect algebras. Adv. Appl. Math. 19, 200–215 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cattaneo, G.: A unified framework for the algebra of unsharp quantum mechanics. Int. J. Theor. Phys. 36, 3085–3117 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1959)

    Article  Google Scholar 

  4. Chen, C.C., Grätzer, G.: Stone lattices I. Construction theorems. Can. J. Math. 21, 884–894 (1969)

    MATH  Google Scholar 

  5. Chen, C.C., Grätzer, G.: Stone lattices II. Structure theorems. Can. J. Math. 21, 895–903 (1969)

    MATH  Google Scholar 

  6. Chevalier, G., Pulmannová, S.: Some ideal lattices in partial abelian monoids. Order 17, 75–92 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chovanec, F., Kôpka, F.: Boolean D-posets. Tatra Mt. Math. Publ. 10, 183–197 (1997)

    MATH  MathSciNet  Google Scholar 

  8. Dvurečenskij, A.: Effect algebras which can be covered by MV-algebras. Int. J. Theor. Phys. 41, 221–229 (2002)

    Article  MATH  Google Scholar 

  9. Dvurečenskij, A., Graziano, M.G.: Remarks on representations of minimal clans. Tatra Mt. Math. Publ. 15, 31–53 (1998)

    MATH  MathSciNet  Google Scholar 

  10. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer and Ister Science, Dordrecht (2000)

    MATH  Google Scholar 

  11. Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)

    Article  MathSciNet  Google Scholar 

  12. Foulis, D.J., Greechie, R., Rütimann, G.: Filters and supports in orthoalgebras. Int. J. Theor. Phys. 35, 789–802 (1995)

    Google Scholar 

  13. Giuntini, R., Greuling, H.: Toward a formal language for unsharp properties. Found. Phys. 19, 931–945 (1989)

    Article  MathSciNet  Google Scholar 

  14. Greechie, R., Foulis, D., Pulmannová, S.: The center of an effect algebra. Order 12, 91–106 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gudder, S.: Sharply dominating effect algebras. Tatra Mt. Math. Publ. 15, 23–30 (1998)

    MATH  MathSciNet  Google Scholar 

  16. Jenča, G.: Subcentral ideals in generalized effect algebras. Int. J. Theor. Phys. 39, 745–755 (2000)

    Article  MATH  Google Scholar 

  17. Jenča, G.: Blocks of homogeneous effect algebras. Bull. Aust. Math. Soc. 64, 81–98 (2001)

    Article  MATH  Google Scholar 

  18. Jenča, G.: Finite homogeneous and lattice ordered effect algebras. Discrete Math. 272, 197–214 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jenča, G., Pulmannová, S.: Quotients of partial abelian monoids and the Riesz decomposition property. Algebra Univers. 47, 443–477 (2002)

    Article  MATH  Google Scholar 

  20. Jenča, G., Pulmannová, S.: Orthocomplete effect algebras. Proc. Am. Math. Soc. 131, 2663–2671 (2003)

    Article  MATH  Google Scholar 

  21. Jenča, G., Riečanová, Z.: On sharp elements in lattice ordered effect algebras. BUSEFAL 80, 24–29 (1999)

    Google Scholar 

  22. Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994)

    MATH  MathSciNet  Google Scholar 

  23. Mundici, D.: Interpretation of AF C *-algebras in Lukasziewicz sentential calculus. J. Funct. Anal. 65, 15–53 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mundici, D.: MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Math. Jpn. 6, 889–894 (1986)

    MathSciNet  Google Scholar 

  25. Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991)

    MATH  Google Scholar 

  26. Riečanová, Z.: A generalization of blocks for d-lattices and lattice effect algebras. Int. J. Theor. Phys. 39, 231–237 (2000)

    Article  MATH  Google Scholar 

  27. Riečanová, Z.: Continuous lattice effect algebras admitting order continuous states. Fuzzy Sets Syst. 136, 41–54 (2003)

    Article  MATH  Google Scholar 

  28. Yutani, H.: The class of commutative BCK-algebras is equationally definable. Math. Semin. Notes 5, 207–210 (1977)

    MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Radlinského 11, Bratislava, 813 68, Slovak Republic

    Gejza Jenča

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  1. Gejza Jenča
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Correspondence to Gejza Jenča.

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Jenča, G. Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras. Order 27, 41–61 (2010). https://doi.org/10.1007/s11083-009-9137-5

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