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On compatibility in quantum logics. (English) Zbl 0539.03045
The notions of compatibility of a given subset of the quantum logic L (i.e., the orthomodular $$\sigma$$-orthoposet) are presented and relationships between them are shown. Many modifications of this notion have been known in the literature and the authors here compare them. (i) $$A\subset L$$ is said to be compatible if $$a\leftrightarrow b$$ for any a,$$b\in A$$; (ii) L satisfies the condition (c) if, for any a,b,$$c\in L$$ which are mutually compatible, we have $$a\leftrightarrow b\vee c;$$ (iii) $$A\subset L$$ is strongly compatible if, for any a,$$b\in A$$, $$a\leftrightarrow b$$ and $$a=a_ 1\oplus c$$, $$b=b_ 1\oplus c$$, we have $$a_ 1,b_ 1,c\in L_ 0(A)$$, where $$L_ 0(A)$$ is the minimal sublogic of L generated by A; (iv) $$A\subset L$$ is full compatible if, for any finite collection $$a_ 1,...,a_ n\in A,$$ there is a finite subset of pairwise orthogonal elements, $$\{e_ i\}$$, such that any $$a_ i=\bigvee_{j}e_{i_ j}$$ for some subcollection of $$\{e_ i\}$$. These notions are applied to the problem of the existence of the minimal sub-$$\sigma$$-algebra of L generated by A, and to the existence of the joint distribution of a given system of observables in a state.
Reviewer: A.Dvurečenskij

MSC:
 03G12 Quantum logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 81P20 Stochastic mechanics (including stochastic electrodynamics)