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Keywords:
orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics
orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics
Summary:
It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$. Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$. This result is "best possible" - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$, then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.
References:
[1] V. Alda: On 0-1 measures for projectors. Aplikace Matematiky 26, 57-58 (1981). MR 0602402
[2] L. J. Bunce D. M. Wright: Qantum measures and states on Jordan algebras. Comm. Math. Phys. (To appear). MR 0786572
[3] J. Brabec P. Pták: On compatibility in quantum logics. Foundations of Physics, Vol. 12, No. 2, 207-212 (1982). DOI 10.1007/BF00736849 | MR 0659779
[4] R. Godowski: Varieties of orthomodular lattices with a strongly full set of states. Demonstration Mathematica, Vol. XIV, No. 3, (1981). MR 0663122 | Zbl 0483.06007
[5] R. Greechie: Orthomodular lattices admitting no states. J. Comb. Theory 10, 119-132 (1971). DOI 10.1016/0097-3165(71)90015-X | MR 0274355 | Zbl 0219.06007
[6] S. Gudder: Stochastic Methods in Quantum Mechanics. North-Holland 1979. MR 0543489 | Zbl 0439.46047
[7] P. Pták: Weak dispersion-free states and the hidden variables hypothesis. J. Math. Physics 24 (4), 839-840(1983). DOI 10.1063/1.525758 | MR 0700618
[8] P. Pták V. Rogolewicz: Measures on orthomodular partially ordered sets. J. Pure and Applied Algebra 28, 75-85 (1983). DOI 10.1016/0022-4049(83)90074-9 | MR 0692854
[9] S. Pulmannová: Compatibility and partial compatibility in quantum logics. Ann. Inst. Henri Poincaré, Vol. XXXIV, No. 4, 391-403 (1981). MR 0625170
[10] R. Sikorski: Boolean Algebras. Springer-Verlag (1964). MR 0126393 | Zbl 0123.01303
[11] V. Varadarajan: Geometry of Quantum Theory I. Von Nostrand, Princeton (1968). MR 0471674
[12] M. Zierler M. Schlessinger: Boolean embedding of orthomodular sets and quantum logics. Duke J. Math. 32, 251-262 (1965). DOI 10.1215/S0012-7094-65-03224-2 | MR 0175520
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