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Basic decomposition of elements and Jauch-Piron effect algebras. (English) Zbl 1073.81014
Summary: We show that every element of a complete atomic effect algebra \(E\) has a unique basic decomposition into a sum of a sharp element and unsharp multiples of isotropic atoms of \(E\). Consequently, for such effect algebras we obtain “the Smearing Theorem for states” establishing that every order-continuous state existing on sharp elements of \(E\) can be extended to a state on \(E\). For a \(\sigma\)-complete separable atomic effect algebra \(E\) we prove that E is a unital and Jauch-Piron effect algebra if and only if the set \(S(E)\) of all sharp elements of \(E\) is a unital Jauch-Piron orthomodular lattice and for finite \(E\), \(S(E)\) is a Boolean algebra.

MSC:
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
06C15 Complemented lattices, orthocomplemented lattices and posets
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