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On ring-like structures induced by Mackey’s probability function. (English) Zbl 1056.81004
Summary: The aim of this paper is to show that the structure of the set $$L$$ of experimental propositions induced by Mackey’s probability function $$p(A, \alpha, E)$$ can be conveniently described and investigated in the framework of the theory of ring-like structures called generalized Boolean quasirings. With the help of only one axiom (the so-called extension axiom) a representation theorem for $$L$$ is stated and proved. The operations $$\cdot$$ and $$+$$ generalizing the classical “and”-operation and the classical “exclusive or”-operation, respectively, are investigated. It is shown that under some natural assumptions the associativity of $$+$$ implies the distributivity of $$L$$ and that the unique solvability of the equations $$a+X=b$$ is equivalent to the classicality of $$L$$. A physical interpretation for the obtained results is given.

##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G12 Quantum logic 06C15 Complemented lattices, orthocomplemented lattices and posets
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##### References:
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