Lattice Properties of Ring-like Quantum Logics |
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Authors: | Dietmar Dorninger Helmut Länger Maciej Maczyński |
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Affiliation: | (1) Institut fü Algebra und Computermathematik, Technische Universitä Wien, A-1040 Vienna, Austria;(2) Instytut Matematyki, Politechnika Warszawska, PL 00-661 Warsaw, Poland |
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Abstract: | Generalized Boolean quasirings (GBQRs) are extensions of partial algebras thatare in one-to-one correspondence to bounded lattices with an involutoryantiautomorphism. This correspondence generalizes the bijection betweenBoolean rings and Boolean algebras and provides for a large variety of presumptivequantum logics (including logics which can be defined by means of Mackey'sprobability function). It is shown how properties of the corresponding latticesare reflected in GBQRs and what the implications are of the associativity of the+-operation of GBQRs, which can be interpreted as some kind of an exclusiveor-operation. We prove that under very weak conditions, which, however, seemto be essential for experimental verifications, the associativity of + implies theclassicality of the considered quantum mechanical system. |
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