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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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