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On central atoms of Archimedean atomic lattice effect algebras. (English) Zbl 1214.06002
Let $$E$$ be an atomic orthomodular lattice. J. Paseka and Z. Riečanová asked whether the center of $$E$$ is a bifull sublattice of $$E$$ [J. Paseka and Z. Riečanová, “The inheritance of BDE-property in sharply dominating lattice effect algebras and ($$o$$)-continuous states”, Soft Comput. 15, No. 3, 543–555 (2011)]. The author constructs a counterexample.
From previous work, it is known that in such a counterexample the supremum of all atoms in the center of $$E$$ cannot be the top element (even if $$E$$ is an Archimedean atomic lattice effect algebra, see [Z. Riečanová, “Subdirect decompositions of lattice effect algebras”, Int. J. Theor. Phys. 42, No. 7, 1425–1433 (2003; Zbl 1034.81003)]). Thus, the result requires several non-trivial tricks. Moreover, the orthomodular lattice constructed here is representable by subsets of a set (i.e., it has an order-determining set of two-valued states). It is a clever construction contributing to our knowledge of quantum structures.
There is a continuation of this paper by the author [M. Kalina, “Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras”, Kybernetika 46, No. 6, 935–947 (2010; Zbl 1221.06010)].

##### MSC:
 06C15 Complemented lattices, orthocomplemented lattices and posets 03G12 Quantum logic 06D35 MV-algebras
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##### References:
 [1] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad. Publisher, Dordrecht, Boston, London, and Isterscience, Bratislava 2000. · Zbl 0987.81005 [2] Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325-1346. · Zbl 1213.06004 · doi:10.1007/BF02283036 [3] Greechie, R. J., Foulis, D. J., Pulmannová, S.: The center of an effect algebra. Order 12 (1995), 91-106. · Zbl 0846.03031 · doi:10.1007/BF01108592 [4] Gudder, S. P.: Sharply dominating effect algebras, Tatra Mountains Math. Publ. 15 (1998), 23-30. · Zbl 0939.03073 [5] Gudder, S. P.: S-dominating effect algebras. Internat. J. Theor. Phys. 37 (1998), 915-923. · Zbl 0932.03072 · doi:10.1023/A:1026637001130 [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. · Zbl 0843.03042 · doi:10.1007/BF00676263 [8] Mosná, K.: About atoms in generalized efect algebras and their effect algebraic extensions. J. Electr. Engrg. 57 (2006), 7/s, 110-113. · Zbl 1118.03060 [9] Mosná, K., Paseka, J., Riečanová, Z.: Order convergence and order and interval topologies on posets and lattice effect algebras. UNCERTAINTY2008, Proc. Internat. Seminar, Publishing House of STU 2008, pp. 45-62. [10] Paseka, J., Riečanová, Z.: The inheritance of BDE-property in sharply dominating lattice effect algebras and $$(o)$$-continuous states. Soft Computing, to appear. · Zbl 1247.03135 [11] Riečanová, Z.: Compatibility and central elements in effect algebras. Tatra Mountains Math. Publ. 16 (1999), 151-158. · Zbl 0949.03063 [12] Riečanová, Z.: Subalgebras, intervals and central elements of generalized effect algebras. Internat. J. Theor. Phys. 38 (1999), 3209-3220. · Zbl 0963.03087 · doi:10.1023/A:1026682215765 [13] Riečanová, Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras Internat. J. Theor. Phys. 39 (2000), 231-237. · Zbl 0968.81003 · doi:10.1023/A:1003619806024 [14] Riečanová, Z.: Orthogonal sets in effect algebras. Demonstratio Math. 34 (2001), 525-532. · Zbl 0989.03071 [15] Riečanová, Z.: Smearing of states defined on sharp elements onto effect algebras. Interernat. J. Theor. Phys. 41 (2002), 1511-1524. · Zbl 1016.81005 · doi:10.1023/A:1020136531601 [16] Riečanová, Z.: Subdirect decompositions of lattice effect algebras. Interernat. J. Theor. Phys. 42 (2003), 1425-1433. · Zbl 1034.81003 · doi:10.1023/A:1025775827938 [17] Riečanová, Z.: Distributive atomic effect akgebras. Demonstratio Math. 36 (2003), 247-259. · Zbl 1039.03050 [18] Riečanová, Z., Marinová, I.: Generalized homogenous, prelattice and MV-effect algebras. Kybernetika 41 (2005), 129-142. · Zbl 1249.03122 · www.kybernetika.cz
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