Geodesic
Orthocomplemented difference lattices with few generators
Kybernetika, Tome 47 (2011) no. 1, pp. 60-73 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 ${\rm MO}_3 \times 2^4$.
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 ${\rm MO}_3 \times 2^4$.
Classification : 03G12, 06C15, 81B10
Keywords: orthomodular lattice; quantum logic; symmetric difference; Gödel's coding; Boolean algebra; free algebra 
@article{KYB_2011_47_1_a4,
     author = {Matou\v{s}ek, Milan and Pt\'ak, Pavel},
     title = {Orthocomplemented difference lattices with few generators},
     journal = {Kybernetika},
     pages = {60--73},
     year = {2011},
     volume = {47},
     number = {1},
     mrnumber = {2807864},
     zbl = {1221.06011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2011_47_1_a4/}
}
TY  - JOUR
AU  - Matoušek, Milan
AU  - Pták, Pavel
TI  - Orthocomplemented difference lattices with few generators
JO  - Kybernetika
PY  - 2011
SP  - 60
EP  - 73
VL  - 47
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/KYB_2011_47_1_a4/
LA  - en
ID  - KYB_2011_47_1_a4
ER  - 
%0 Journal Article
%A Matoušek, Milan
%A Pták, Pavel
%T Orthocomplemented difference lattices with few generators
%J Kybernetika
%D 2011
%P 60-73
%V 47
%N 1
%U http://geodesic.mathdoc.fr/item/KYB_2011_47_1_a4/
%G en
%F KYB_2011_47_1_a4
Matoušek, Milan; Pták, Pavel. Orthocomplemented difference lattices with few generators. Kybernetika, Tome 47 (2011) no. 1, pp. 60-73. http://geodesic.mathdoc.fr/item/KYB_2011_47_1_a4/

[1] Beran, L.: Orthomodular Lattices, Algebraic Approach. D. Reidel, Dordrecht, 1985. | MR | Zbl

[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. | MR | Zbl

[3] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. Springer-Verlag, New York 1981. | MR | Zbl

[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

[5] Dorfer, G., Dvurečenskij, A., Länger, H. M.: Symmetric difference in orthomodular lattices. Math. Slovaca 46 (1996), 435–444. | MR

[6] Dorfer, G.: Non-commutative symmetric differences in orthomodular lattices. Internat. J. Theoret. Phys. 44 (2005), 885–896. | DOI | MR

[7] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, and Ister Science, Bratislava 2000. | MR

[8] Godowski, R., Greechie, R. J.: Some equations related to states on orthomodular lattices. Demonstratio Math. XVII (1984), 1, 241–250. | MR | Zbl

[9] Greechie, R. J.: Orthomodular lattices admitting no states. J. Combinat. Theory 10 (1971), 119–132. | DOI | MR | Zbl

[10] Engesser, K., Gabbay, D.M., ed., D. Lehmann: Handbook of Quantum Logic and Quantum Structures. Elsevier 2007. | MR

[11] Havlík, F.: Ortokomplementární diferenční svazy (in Czech). (Mgr. Thesis, Department of Logic, Faculty of Arts, Charles University in Prague 2007).

[12] Kalmbach, G.: Orthomodular Lattices. Academic Press, London 1983. | MR | Zbl

[13] Matoušek, M.: Orthocomplemented lattices with a symmetric difference. Algebra Universalis 60 (2009), 185–215. | DOI | MR | Zbl

[14] Matoušek, M., Pták, P.: Orthocomplemented posets with a symmetric difference. Order 26 (2009), 1–21. | DOI | MR | Zbl

[15] Matoušek, M., Pták, P.: On identities in orthocomplemented difference lattices. Math. Slovaca 60 (2010), 5, 583–590. | DOI | MR | Zbl

[16] Matoušek, M., Pták, P.: Symmetric difference on orthomodular lattices and $Z_2$-valued states. Comment. Math. Univ. Carolin. 50 (2009), 4, 535–547. | MR | Zbl

[17] Navara, M., Pták, P.: Almost Boolean orthomodular posets. J. Pure Appl. Algebra 60 (1989), 105–111. | DOI | MR

[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. | DOI | MR | Zbl

[19] Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht, Boston, London 1991. | MR

[20] Watanabe, S.: Modified concepts of logic, probability and integration based on generalized continuous characteristic function. Inform. and Control 2 (1969), 1–21. | DOI | MR