Automorphisms of concrete logics

Mirko Navara; Josef Tkadlec

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 1, page 15-25
  • ISSN: 0010-2628

Abstract

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The main result of this paper is Theorem 3.3: Every concrete logic (i.e., every set-representable orthomodular poset) can be enlarged to a concrete logic with a given automorphism group and with a given center. Since every sublogic of a concrete logic is concrete, too, and since not every state space of a (general) quantum logic is affinely homeomorphic to the state space of a concrete logic [8], our result seems in a sense the best possible. Further, we show that every group is an automorphism group of a concrete lattice logic and, on the other hand, we prove that this is not true for Boolean logics with a dense center. As a technical tool for pursuing the latter type of problems, we investigate the correspondence between homomorphisms of concrete logics and pointwise mappings of their domain. We prove that in a suitable topological representation of concrete logics, every automorphism is carried by a homeomorphism.

How to cite

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Navara, Mirko, and Tkadlec, Josef. "Automorphisms of concrete logics." Commentationes Mathematicae Universitatis Carolinae 32.1 (1991): 15-25. <http://eudml.org/doc/247253>.

@article{Navara1991,
abstract = {The main result of this paper is Theorem 3.3: Every concrete logic (i.e., every set-representable orthomodular poset) can be enlarged to a concrete logic with a given automorphism group and with a given center. Since every sublogic of a concrete logic is concrete, too, and since not every state space of a (general) quantum logic is affinely homeomorphic to the state space of a concrete logic [8], our result seems in a sense the best possible. Further, we show that every group is an automorphism group of a concrete lattice logic and, on the other hand, we prove that this is not true for Boolean logics with a dense center. As a technical tool for pursuing the latter type of problems, we investigate the correspondence between homomorphisms of concrete logics and pointwise mappings of their domain. We prove that in a suitable topological representation of concrete logics, every automorphism is carried by a homeomorphism.},
author = {Navara, Mirko, Tkadlec, Josef},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {orthomodular lattice; quantum logic; concrete logic; set representation; automorphism group of a logic; state space; set-representable orthomodular poset; concrete quantum logic; automorphism group; concrete orthomodular lattice},
language = {eng},
number = {1},
pages = {15-25},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Automorphisms of concrete logics},
url = {http://eudml.org/doc/247253},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Navara, Mirko
AU - Tkadlec, Josef
TI - Automorphisms of concrete logics
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 1
SP - 15
EP - 25
AB - The main result of this paper is Theorem 3.3: Every concrete logic (i.e., every set-representable orthomodular poset) can be enlarged to a concrete logic with a given automorphism group and with a given center. Since every sublogic of a concrete logic is concrete, too, and since not every state space of a (general) quantum logic is affinely homeomorphic to the state space of a concrete logic [8], our result seems in a sense the best possible. Further, we show that every group is an automorphism group of a concrete lattice logic and, on the other hand, we prove that this is not true for Boolean logics with a dense center. As a technical tool for pursuing the latter type of problems, we investigate the correspondence between homomorphisms of concrete logics and pointwise mappings of their domain. We prove that in a suitable topological representation of concrete logics, every automorphism is carried by a homeomorphism.
LA - eng
KW - orthomodular lattice; quantum logic; concrete logic; set representation; automorphism group of a logic; state space; set-representable orthomodular poset; concrete quantum logic; automorphism group; concrete orthomodular lattice
UR - http://eudml.org/doc/247253
ER -

References

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  1. Groot J. de, McDowell R.H., Autohomeomorphism groups of 0-dimensional spaces, Compositio Math. 15 (1963), 203-209. (1963) MR0154267
  2. Gudder S.P., Stochastic Methods in Quantum Mechanics, North Holland, New York, 1979. Zbl0439.46047MR0543489
  3. Iturrioz L., A representation theory for orthomodular lattices by means of closure spaces, Acta Math. Hung. 47 (1986), 145-151. (1986) Zbl0608.06008MR0836405
  4. Kallus M., Trnková V., Symmetries and retracts of quantum logics, Int. J. Theor. Phys. 26 (1987), 1-9. (1987) MR0890206
  5. Kalmbach G., Orthomodular Lattices, Academic Press, London, 1983. Zbl0554.06009MR0716496
  6. Kalmbach G., Automorphism groups of orthomodular lattices, Bull. Austral. Math. Soc. 29 (1984), 309-313. (1984) Zbl0538.06009MR0748724
  7. Navara M., The independence of automorphism groups, centres and state spaces in quantum logics, to appear. 
  8. Navara M., An alternative proof of Shultz's theorem, to appear. 
  9. Navara M., Pták P., Almost Boolean orthomodular posets, J. Pure Appl. Algebra 60 (1989), 105-111. (1989) MR1014608
  10. Navara M., Rogalewicz V., The pasting constructions for orthomodular posets, to appear in Math. Nachrichten. Zbl0767.06009MR1138377
  11. Pták P., Logics with given centres and state spaces, Proc. Amer. Math. Soc. 88 (1983), 106-109. (1983) MR0691287
  12. Pták P., Pulmannová S., Orthomodular Structures as Quantum Logics, Kluwer, 1991 (to appear). MR1176314
  13. Pultr A., Trnková V., Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, North-Holland, Amsterdam, 1980 and Academia, Praha, 1980. MR0563525
  14. Sikorski R., Boolean Algebras, Springer-Verlag, Berlin, 1969. Zbl0191.31505MR0126393
  15. Tkadlec J., Set representations of orthoposets, Proc. 2-nd Winter School on Measure Theory (Liptovský Ján, 1990), Slovak Academy of Sciences, Bratislava, 1990. Zbl0777.06009MR1118430
  16. Zierler N., Schlessinger M., Boolean embeddings of orthomodular sets and quantum logic, Duke Math. J. 32 (1965), 251-262. (1965) MR0175520

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