| Title:
|
Orthocomplemented difference lattices with few generators (English) |
| Author:
|
Matoušek, Milan |
| Author:
|
Pták, Pavel |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
47 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
60-73 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e. g., [19, 20]). Recently an effort has been exercised to advance with logics that possess a symmetric difference ([13, 14]) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In [13] the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result of this paper somewhat economizes on this construction: There is an ODL with 3 generators that is not set-representable (and so the free ODL with 3 generators cannot be set-representable). The result is based on a specific technique of embedding orthomodular lattices into ODLs. The ODLs with 2 generators are always set-representable as we show by characterizing the free ODL with 2 generators - this ODL is ${\rm MO}_3 \times 2^4$. (English) |
| Keyword:
|
orthomodular lattice |
| Keyword:
|
quantum logic |
| Keyword:
|
symmetric difference |
| Keyword:
|
Gödel's coding |
| Keyword:
|
Boolean algebra |
| Keyword:
|
free algebra |
| MSC:
|
03G12 |
| MSC:
|
06C15 |
| MSC:
|
81B10 |
| idZBL:
|
Zbl 1221.06011 |
| idMR:
|
MR2807864 |
| . |
| Date available:
|
2011-04-12T13:03:47Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141478 |
| . |
| Reference:
|
[1] Beran, L.: Orthomodular Lattices, Algebraic Approach.D. Reidel, Dordrecht, 1985. Zbl 0558.06008, MR 0784029 |
| Reference:
|
[2] Bruns, G., Harding, J.: Algebraic aspects of orthomodular lattices.In: Current Research in Operational Quantum Logic (B. Coecke, D. Moore and A. Wilce, eds.), Kluwer Academic Publishers 2000, pp. 37–65. Zbl 0955.06003, MR 1907155 |
| Reference:
|
[3] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra.Springer-Verlag, New York 1981. Zbl 0478.08001, MR 0648287 |
| Reference:
|
[4] Chiara, M. L. Dalla, Giuntini, R., Greechie, R.: Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics.Kluwer Academic Publishers, Dordrecht, Boston, London 2004. MR 2069854 |
| Reference:
|
[5] Dorfer, G., Dvurečenskij, A., Länger, H. M.: Symmetric difference in orthomodular lattices.Math. Slovaca 46 (1996), 435–444. MR 1451034 |
| Reference:
|
[6] Dorfer, G.: Non-commutative symmetric differences in orthomodular lattices.Internat. J. Theoret. Phys. 44 (2005), 885–896. MR 2199505, 10.1007/s10773-005-7066-7 |
| Reference:
|
[7] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures.Kluwer Acad. Publ., Dordrecht, and Ister Science, Bratislava 2000. MR 1861369 |
| Reference:
|
[8] Godowski, R., Greechie, R. J.: Some equations related to states on orthomodular lattices.Demonstratio Math. XVII (1984), 1, 241–250. Zbl 0553.06013, MR 0760356 |
| Reference:
|
[9] Greechie, R. J.: Orthomodular lattices admitting no states.J. Combinat. Theory 10 (1971), 119–132. Zbl 0219.06007, MR 0274355, 10.1016/0097-3165(71)90015-X |
| Reference:
|
[10] Engesser, K., Gabbay, D.M., ed., D. Lehmann: Handbook of Quantum Logic and Quantum Structures.Elsevier 2007. MR 2408886 |
| Reference:
|
[11] Havlík, F.: Ortokomplementární diferenční svazy (in Czech).(Mgr. Thesis, Department of Logic, Faculty of Arts, Charles University in Prague 2007). |
| Reference:
|
[12] Kalmbach, G.: Orthomodular Lattices.Academic Press, London 1983. Zbl 0528.06012, MR 0716496 |
| Reference:
|
[13] Matoušek, M.: Orthocomplemented lattices with a symmetric difference.Algebra Universalis 60 (2009), 185–215. Zbl 1186.06004, MR 2491422, 10.1007/s00012-009-2105-5 |
| Reference:
|
[14] Matoušek, M., Pták, P.: Orthocomplemented posets with a symmetric difference.Order 26 (2009), 1–21. Zbl 1201.06006, MR 2487165, 10.1007/s11083-008-9102-8 |
| Reference:
|
[15] Matoušek, M., Pták, P.: On identities in orthocomplemented difference lattices.Math. Slovaca 60 (2010), 5, 583–590. Zbl 1249.06025, MR 2728524, 10.2478/s12175-010-0033-7 |
| Reference:
|
[16] Matoušek, M., Pták, P.: Symmetric difference on orthomodular lattices and $Z_2$-valued states.Comment. Math. Univ. Carolin. 50 (2009), 4, 535–547. Zbl 1212.06021, MR 2583131 |
| Reference:
|
[17] Navara, M., Pták, P.: Almost Boolean orthomodular posets.J. Pure Appl. Algebra 60 (1989), 105–111. MR 1014608, 10.1016/0022-4049(89)90108-4 |
| Reference:
|
[18] Park, E., Kim, M. M., Chung, J. Y.: A note on symmetric differences of orthomodular lattices.Commun. Korean Math. Soc. 18 (2003), 2, 207–214. Zbl 1101.06301, MR 1986740, 10.4134/CKMS.2003.18.2.207 |
| Reference:
|
[19] Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics.Kluwer Academic Publishers, Dordrecht, Boston, London 1991. MR 1176314 |
| Reference:
|
[20] Watanabe, S.: Modified concepts of logic, probability and integration based on generalized continuous characteristic function.Inform. and Control 2 (1969), 1–21. MR 0267964, 10.1016/S0019-9958(69)90581-6 |
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