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lattice effect algebra; center; atom; bifullness
If element $z$ of a lattice effect algebra $(E,\oplus, {\mathbf 0}, {\mathbf 1})$ is central, then the interval $[{\mathbf 0},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$. It is a known fact that if the set $C(E)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C(E)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C(E)$ is a bifull sublattice of an Archimedean atomic lattice effect algebra $E$. We show that there exists a lattice effect algebra $(E,\oplus, {\mathbf 0}, {\mathbf 1})$ with atomic $C(E)$ which is not a bifull sublattice of $E$. Moreover, we show that also $B(E)$, the center of compatibility, may not be a bifull sublattice of $E$.
[1] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad. Publisher, Dordrecht, Boston, London, and Isterscience, Bratislava 2000. MR 1861369
[2] Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325–1346. MR 1304942
[3] Greechie, R. J., Foulis, D. J., Pulmannová, S.: The center of an effect algebra. Order 12 (1995), 91–106. DOI 10.1007/BF01108592 | MR 1336539
[4] Gudder, S. P.: Sharply dominating effect algebras, Tatra Mountains Math. Publ. 15 (1998), 23–30. MR 1655076
[5] Gudder, S. P.: S-dominating effect algebras. Internat. J. Theor. Phys. 37 (1998), 915–923. DOI 10.1023/A:1026637001130 | MR 1624277 | Zbl 0932.03072
[6] Jenča, G., Riečanová, Z.: On sharp elements in lattice ordered effect algebras. BUSEFAL 80 (1999), 24–29.
[7] Kôpka, F.: Compatibility in D-posets. Interernat. J. Theor. Phys. 34 (1995), 1525–1531. DOI 10.1007/BF00676263 | MR 1353696
[8] Mosná, K.: About atoms in generalized efect algebras and their effect algebraic extensions. J. Electr. Engrg. 57 (2006), 7/s, 110–113.
[9] Mosná, K., Paseka, J., Riečanová, Z.: Order convergence and order and interval topologies on posets and lattice effect algebras. In: UNCERTAINTY2008, Proc. Internat. Seminar, Publishing House of STU 2008, pp. 45–62. MR 2395159
[10] Paseka, J., Riečanová, Z.: The inheritance of BDE-property in sharply dominating lattice effect algebras and $(o)$-continuous states. Soft Computing, to appear.
[11] Riečanová, Z.: Compatibility and central elements in effect algebras. Tatra Mountains Math. Publ. 16 (1999), 151–158. MR 1725293
[12] Riečanová, Z.: Subalgebras, intervals and central elements of generalized effect algebras. Internat. J. Theor. Phys. 38 (1999), 3209–3220. DOI 10.1023/A:1026682215765 | MR 1764459
[13] Riečanová, Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras Internat. J. Theor. Phys. 39 (2000), 231–237. DOI 10.1023/A:1003619806024 | MR 1762594
[14] Riečanová, Z.: Orthogonal sets in effect algebras. Demonstratio Math. 34 (2001), 525–532. MR 1853730 | Zbl 0989.03071
[15] Riečanová, Z.: Smearing of states defined on sharp elements onto effect algebras. Interernat. J. Theor. Phys. 41 (2002), 1511–1524. DOI 10.1023/A:1020136531601 | MR 1932844
[16] Riečanová, Z.: Subdirect decompositions of lattice effect algebras. Interernat. J. Theor. Phys. 42 (2003), 1425–1433. DOI 10.1023/A:1025775827938 | MR 2021221 | Zbl 1034.81003
[17] Riečanová, Z.: Distributive atomic effect akgebras. Demonstratio Math. 36 (2003), 247–259. MR 1984337
[18] Riečanová, Z., Marinová, I.: Generalized homogenous, prelattice and MV-effect algebras. Kybernetika 41 (2005), 129–142. MR 2138764
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