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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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