| Title:
|
Descriptions of state spaces of orthomodular lattices (the hypergraph approach) (English) |
| Author:
|
Navara, Mirko |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
117 |
| Issue:
|
3 |
| Year:
|
1992 |
| Pages:
|
305-313 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Using the general hypergraph technique developed in [7], we first give a much simpler proof of Shultz's theorem [10]: Each compact convex set is affinely homeomorphic to the state space of an orthomodular lattice. We also present partial solutions to open questions formulated in [10] - we show that not every compact convex set has to be a state space of a unital orthomodular lattice and that for unital orthomodular lattices the state space characterization can be obtained in the context of unital hypergraphs. (English) |
| Keyword:
|
affine homeomorphism |
| Keyword:
|
compact convex set |
| Keyword:
|
hypergraph |
| Keyword:
|
unital orthomodular lattices |
| Keyword:
|
state space representation |
| Keyword:
|
orthomodular lattice |
| Keyword:
|
state space |
| MSC:
|
03G12 |
| MSC:
|
05C65 |
| MSC:
|
06C15 |
| idZBL:
|
Zbl 0772.06008 |
| idMR:
|
MR1184544 |
| DOI:
|
10.21136/MB.1992.126280 |
| . |
| Date available:
|
2009-09-24T20:54:05Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126280 |
| . |
| Reference:
|
[1] Greechie R.J.: Orthomodular lattices admitting no states.J. Comb. Theory 10(1971), 119-132. Zbl 0219.06007, MR 0274355, 10.1016/0097-3165(71)90015-X |
| Reference:
|
[2] Gudder S.P.: Stochastic Methods in Quantum Mechanics.North Holland, New York, 1979. Zbl 0439.46047, MR 0543489 |
| Reference:
|
[3] Gudder S., Kläy M.P., Rüttimann G.T.: States on hypergraphs.Demonstratio Math. 19 (1986), 503-526. MR 0895021 |
| Reference:
|
[4] Kalmbach G.: Orthomodular Lattices.Academic Press, London, 1983. Zbl 0528.06012, MR 0716496 |
| Reference:
|
[5] Navara M.: State space properties of finite logics.Czechoslovak Math. J. 37 (112) (1987), 188-196. Zbl 0647.03057, MR 0882593 |
| Reference:
|
[6] Navara M.: State space of quantum logics.Thesis, Technical University of Prague, 1987. (In Czech.) |
| Reference:
|
[7] Navara M., Rogatewicz V.: Construction of orthomodular lattices with given state spaces.Demonstratio Math. 21 (1988), 481-493. MR 0981700, 10.1515/dema-1988-0218 |
| Reference:
|
[8] Pták P.: Exotic logics.Coll. Math. 54 (1987), 1-7. MR 0928651, 10.4064/cm-54-1-1-7 |
| Reference:
|
[9] Pták P., Pulmannová S.: Orthomodular Structures as Quantum Logics.Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. Zbl 0743.03039, MR 1176314 |
| Reference:
|
[10] Shultz F. W.: A characterization of state spaces of orthomodular lattices.J. Comb. Theory (A) 17 (1974), 317-328. MR 0364042, 10.1016/0097-3165(74)90096-X |
| . |
