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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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