Pulmannová, Sylvia On representations of logics. (English) Zbl 0531.03040 Math. Slovaca 33, 357-362 (1983). 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 Cited in 1 Document MSC: 03G12 Quantum logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03B60 Other nonclassical logic Keywords:quantum logic; orthomodular orthoposet; representation; Hermitian form Citations:Zbl 0453.03066 PDF BibTeX XML Cite \textit{S. Pulmannová}, Math. Slovaca 33, 357--362 (1983; Zbl 0531.03040) Full Text: EuDML 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 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.