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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.
MSC:
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
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