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On representations of logics. (English) Zbl 0531.03040
This paper continues an earlier paper by the author [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 32, 351-360 (1980; Zbl 0453.03066)] in which an embedding of a logic L (i.e. a special orthomodular orthoposet) into the lattice of all f-closed subspaces \(L_ f(V)\) of a vector space V with the Hermitian form f was found. In the present paper the author proves that \(L_ f(V)\) has the Hilbert property \((M+M^{\perp}=V\) for all \(M\in L_ f(V)\) iff the supremum \(a\vee e\) exists in L for any \(a\in L\) and any atom \(e\in L\).
Reviewer: L.Esakia

MSC:
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03B60 Other nonclassical logic
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References:
[1] BUGAISKA K., BUGAJSKI S.: The lattice structure of quantum logics. Ann. Inst. H. Poincaré 19, 1973, 333-340.
[2] GUDDER S. P.: A superposition principle in physics. J. Math. Phys. 11, 1970, 1037-1040. · Zbl 0191.26901
[3] GUDDER S. P.: Projective representations of quantum logics. Int. J. Theoret. Phys. 3, 1970, 99-108.
[4] GUZ W.: On the lattice structure of quantum logics. Ann. Inst. H. Poincaré 28, 1978, 1-7.
[5] GUZ W.: Filter theory and covering law. Ann. Inst. H. Poincaré 29, 1978, 357-378. · Zbl 0398.03052
[6] JAUCH J.: Foundations of Quantum Mechanics. Addison-Wesley Publ. Co., Inc. Reading, Mass. 1968. · Zbl 0166.23301
[7] MAC LAREN M. D.: Atomic orthocomplemented lattices. Pac. J. Math. 14, 1964, 597-612. · Zbl 0122.02201
[8] MAEDA F., MAEDA S.: Theory of Symmetric Lattices. Springer-Verlag, Berlin, 1970. · Zbl 0219.06002
[9] PIRON C.: Foundations of Quantum Physics. W. A. Benjamin, Inc. Reading, Mass. 1967. · Zbl 0333.46050
[10] PULMANNOVÁ S.: A superposition principle in quantum logics. Commun. Math. Phys. 49, 1976, 47-51.
[11] PULMANNOVÁ S.: The superposition principle and sectors in quantum logics. Int. J. Theor. Phys. 18, 1979, 915-920. · Zbl 0491.03022
[12] PULMANNOVÁ S.: Superpositions of states and a representation theorem. Ann. Inst. H. Poincaré 32, 1980, 351-360. · Zbl 0453.03066
[13] VARADARAJAN V. S.: Geometry of Quantum Theory. Van Nostrand, Pnnceton N.Y. 1968. · Zbl 0155.56802
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