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Extensions of ordering sets of states from effect algebras onto their MacNeille completions. (English) Zbl 1270.06005
Summary: In [Z. Riečanová and M. Zajac, Rep. Math. Phys. 70, No. 3, 283–290 (2012; Zbl 1268.81014)] it was shown that an effect algebra \(E\) with an ordering set \(\mathcal{M}\) of states can by embedded into a Hilbert space effect algebra \(\mathcal{E}(l_{2}(\mathcal{M}))\). We consider the problem when its effect-algebraic MacNeille completion \(\hat{E}\) can be also embedded into the same Hilbert space effect algebra \(\mathcal {E}(l_{2}(\mathcal{M}))\). That is, when the ordering set \(\mathcal{M}\) of states on \(E\) can be extended to an ordering set of states on \(\hat{E}\). We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.

MSC:
06D35 MV-algebras
06A15 Galois correspondences, closure operators (in relation to ordered sets)
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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