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The sum of observables on a $$\sigma$$-distributive lattice effect algebra. (English) Zbl 1418.03181
Summary: Observables on quantum structures can be seen as generalizations of random variables on a measurable space $$(\Omega , \mathcal {A})$$ for the case when $$\mathcal {A}$$ is not necessarily a Boolean algebra. The present paper investigates an extending of the usual pointwise sum of random variables onto the set of bounded observables on a $$\sigma$$-distributive lattice effect algebra $$E$$. We describe conditions under which this operation, so-called sum $$x+y$$ of observables $$x$$, $$y$$, preserves continuity of spectral resolutions of $$x$$, $$y$$. We show how the spectrum $$\sigma (x+y)$$ depends on spectra $$\sigma (x)$$, $$\sigma (y)$$, and we provide a relation between the meager part $$x_m$$ and the dense part $$x_d$$ of an observable $$x$$.
##### MSC:
 03G12 Quantum logic 06C15 Complemented lattices, orthocomplemented lattices and posets 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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