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Atomicity of lattice effect algebras and their sub-lattice effect algebras. (English) Zbl 1252.06007
Summary: We show some families of lattice effect algebras (a common generalization of orthomodular lattices and MV-effect algebras) each element \(E\) of which has atomic center \(C(E)\) or the subset \(S(E)\) of all sharp elements, respectively the center of compatibility \(B(E)\) or every block \(M\) of \(E\). The atomicity of \(E\) or its sub-lattice effect algebras \(C(E)\), \(S(E)\), \(B(E)\) and blocks \(M\) of \(E\) is a very useful equipment for the investigations of its algebraic and topological properties, like the existence of smearing of states on \(E\), questions about isomorphisms and so on. Namely, we consider the families of complete lattice effect algebras, and lattice effect algebras with finitely many blocks, and complete atomic lattice effect algebra \(E\) with Hausdorff interval topology.
MSC:
06D35 MV-algebras
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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