On the concreteness of quantum logics

Pavel Pták; John David Maitland Wright

Aplikace matematiky (1985)

  • Volume: 30, Issue: 4, page 274-285
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Pták, Pavel, and Wright, John David Maitland. "On the concreteness of quantum logics." Aplikace matematiky 30.4 (1985): 274-285. <http://eudml.org/doc/15405>.

@article{Pták1985,
abstract = {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.},
author = {Pták, Pavel, Wright, John David Maitland},
journal = {Aplikace matematiky},
keywords = {orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics; orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics},
language = {eng},
number = {4},
pages = {274-285},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the concreteness of quantum logics},
url = {http://eudml.org/doc/15405},
volume = {30},
year = {1985},
}

TY - JOUR
AU - Pták, Pavel
AU - Wright, John David Maitland
TI - On the concreteness of quantum logics
JO - Aplikace matematiky
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 30
IS - 4
SP - 274
EP - 285
AB - 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.
LA - eng
KW - orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics; orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics
UR - http://eudml.org/doc/15405
ER -

References

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  2. L. J. Bunce D. M. Wright, Qantum measures and states on Jordan algebras, Comm. Math. Phys. (To appear). MR0786572
  3. J. Brabec P. Pták, 10.1007/BF00736849, Foundations of Physics, Vol. 12, No. 2, 207-212 (1982). (1982) MR0659779DOI10.1007/BF00736849
  4. R. Godowski, Varieties of orthomodular lattices with a strongly full set of states, Demonstration Mathematica, Vol. XIV, No. 3, (1981). (1981) Zbl0483.06007MR0663122
  5. R. Greechie, 10.1016/0097-3165(71)90015-X, J. Comb. Theory 10, 119-132 (1971). (1971) Zbl0219.06007MR0274355DOI10.1016/0097-3165(71)90015-X
  6. S. Gudder, Stochastic Methods in Quantum Mechanics, North-Holland 1979. (1979) Zbl0439.46047MR0543489
  7. P. Pták, 10.1063/1.525758, J. Math. Physics 24 (4), 839-840(1983). (1983) MR0700618DOI10.1063/1.525758
  8. P. Pták V. Rogolewicz, 10.1016/0022-4049(83)90074-9, J. Pure and Applied Algebra 28, 75-85 (1983). (1983) MR0692854DOI10.1016/0022-4049(83)90074-9
  9. S. Pulmannová, Compatibility and partial compatibility in quantum logics, Ann. Inst. Henri Poincaré, Vol. XXXIV, No. 4, 391-403 (1981). (1981) MR0625170
  10. R. Sikorski, Boolean Algebras, Springer-Verlag (1964). (1964) Zbl0123.01303MR0126393
  11. V. Varadarajan, Geometry of Quantum Theory I, Von Nostrand, Princeton (1968). (1968) MR0471674
  12. M. Zierler M. Schlessinger, 10.1215/S0012-7094-65-03224-2, Duke J. Math. 32, 251-262 (1965). (1965) MR0175520DOI10.1215/S0012-7094-65-03224-2

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