Une axiomatisation au premier ordre des arrangements de pseudodroites euclidiennes
Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 883-903.

Nous définissons une structure logique permettant de représenter les classes d’homéomorphismes des arrangements de pseudodroites du plan euclidien. Nous donnons une axiomatisation finie du premier ordre de la réalisabilité des arrangements de pseudodroites.

We define a logical structure making it possible to represent arrangements of pseudolines in the Euclidean plane up to homeomorphism. We give a first-order axiomatisation of realizability of such structures by arrangements.

     author = {Courcelle, Bruno and Olive, Fr\'ed\'eric},
     title = {Une axiomatisation au premier ordre des arrangements de pseudodroites euclidiennes},
     journal = {Annales de l'Institut Fourier},
     pages = {883--903},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {3},
     year = {1999},
     doi = {10.5802/aif.1697},
     zbl = {0973.51006},
     mrnumber = {2000g:52022},
     language = {fr},
     url = {archive.numdam.org/item/AIF_1999__49_3_883_0/}
Courcelle, Bruno; Olive, Frédéric. Une axiomatisation au premier ordre des arrangements de pseudodroites euclidiennes. Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 883-903. doi : 10.5802/aif.1697. http://archive.numdam.org/item/AIF_1999__49_3_883_0/

[1] S.A. Adeleke and P.M. Neumann, Relations related to betweenness: their structure and automorphisms, Memoirs of the Amer. Math. Soc., 623 (1998). | MR 98h:20008 | Zbl 0896.08001

[2] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler, Oriented matroids, Encyclopedia of mathematics and its applications, Vol. 46, Cambridge University Press, 1993. | Zbl 0773.52001

[3] J.E. Goodman, Proof of a conjecture of Burr, Grunbaum and Sloane, Discrete Mathematics, 32 (1980), 27-35. | MR 82b:51005 | Zbl 0444.05029

[4] Goodman, Pseudoline arrangements, In J.E. Goodman and J. O'Rourke, editors, Hanbook of Discrete and Computational Geometry, pages 83-109. CRC Press LLC, 1997. | MR 1730161 | Zbl 0914.51007

[5] J.E. Goodman and R. Pollack, Semispaces of configurations, cell complexes of arrangements, Journal of Combinatorial Theory, Series A, 37 (1984), 257-293. | MR 86e:52011 | Zbl 0551.05002

[6] J.E. Goodman, R. Pollack, R. Wenger, and T. Zamfirescu, Arrangements and topological planes, Amer. Math. Monthly, 101 (1994), 866-878. | MR 95h:51026 | Zbl 0827.51003

[7] B. Grunbaum, Arrangements and spreads. In CBMS Regional Conference, volume 10 of Series in Math. Amer. Math. Soc., Providence, R.I., 1972. | MR 46 #6148 | Zbl 0249.50011

[8] L. Ségoufin and V. Vianu, Spacial databases via topological invariants. Proc. ACM Symp. on Principles of Databases Systems, 1998 (version finale à paraître au J. Comput. Syst. Sciences).

[9] P.W. Shor, Stretchability of pseudolines is NP-hard. In Applied geometry and discrete mathematics, The Victor Klee Festschrift, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 4, 1991, 531-554. | MR 92g:05065 | Zbl 0751.05023