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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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