Dorninger, Dietmar; Länger, Helmut; Maçzyński, Maciej On ring-like structures induced by Mackey’s probability function. (English) Zbl 1056.81004 Rep. Math. Phys. 43, No. 3, 499-515 (1999). 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. Cited in 8 Documents MSC: 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G12 Quantum logic 06C15 Complemented lattices, orthocomplemented lattices and posets × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Beltrametti, E.; Ma̧czyński, M., J. Math. Phys., 32, 1280-1286 (1991) · Zbl 0731.60003 [2] Dorfer, G.; Dvurečenskij, A.; Länger, H., Math. Slovaca, 46, 435-444 (1996) · Zbl 0890.06006 [3] Dorninger, D., Tatra Mt. Math. Publ., 15, 75-83 (1998) · Zbl 0939.03074 [4] Dorninger, D.; Länger, H.; Ma̧czyński, M., Demonstrato Math., 30, 215-232 (1997) · Zbl 0879.06005 [5] Dorninger, D.; Länger, H.; Ma̧czyński, M., Österr. Akad. Wiss. Sitzungsber. Math.-naturw. Kl. Abt., II 206, 279-289 (1997) · Zbl 0945.03095 [6] Kalmbach, G., Orthomodular Lattices (1983), Academic Press: Academic Press London · Zbl 0512.06011 [7] Länger, H., Tatra Mt. Math. Publ., 15, 97-105 (1998) · Zbl 0939.03075 [8] Mackey, G., The Mathematical Foundations of Quantum Mechanics (1963), W. A. Benjamin Inc: W. A. Benjamin Inc Reading, Massachusetts · Zbl 0114.44002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.