zbMATH — the first resource for mathematics

Une axiomatisation au premier ordre des arrangements de pseudodroites euclidiennes. (A first-order axiomatisation of arrangements of Euclidean pseudolines.). (French) Zbl 0973.51006
The authors give an axiomatic description of arrangements of pseudolines which is based on a relation \(\mathbf{cb}(a,b,c,d)\) which may be interpreted as follows: “on the pseudoline \(a\), the intersection of \(a\) with \(c\) lies between the intersection of \(a\) with \(b\) and the intersection of \(a\) with \(d\)”. Altogether there are nine axioms. Six of them describe the properties of the “betweenness relation” \(\mathbf{cb}(a,\cdot ,\cdot ,\cdot)\). One of the axioms may be interpreted as a version of the well-known axiom of Pasch. It is shown that in each arrangement of pseudolines these axioms are satisfied, and on the other hand that each structure satisfying the axioms is isomorphic to an arrangement of pseudolines. An interesting proposition used in the proof describes a condition under which it is possible to enlarge a given arrangement by an additional pseudoline passing through a given finite set of points, one on each pseudoline.
51D20 Combinatorial geometries and geometric closure systems
52C30 Planar arrangements of lines and pseudolines (aspects of discrete geometry)
05B35 Combinatorial aspects of matroids and geometric lattices
Full Text: DOI Numdam EuDML
[1] S.A. ADELEKE and P.M. NEUMANN, Relations related to betweenness: their structure and automorphisms, Memoirs of the Amer. Math. Soc., 623 (1998). · 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. · 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. · 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. · 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. · 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. · 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. · Zbl 0751.05023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.