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Oriented rank three matroids and projective planes. (English) Zbl 0972.52017
Kalhoff shows that classical work about order functions and half-orderings of partial planes by J. Joussen [Result. Math. 4, 55-74 (1981; Zbl 0466.51003)] together with the completion theorem of S. Prieß-Crampe [Math. Z. 99, 305-348 (1967; Zbl 0149.38601)] have strong implications for questions about embedding pseudo-line arrangements (rank 3 oriented matroids) into topological projective planes. In particular, he derives new proofs for the rank 3 case of the Folkman-Lawrence representation theorem for oriented matroids, as well as for completion theorems by J. E. Goodman, R. Pollack, R. Wenger and T. Zamfirescu [Am. Math. Mon. 101, 866-878 (1994; Zbl 0827.51003)].
Kalhoff’s main result generalizes all these results, for which the embeddings depend on the orientations: “For every (not necessarily countable) family of (not necessarily finite, simple) orientable rank three matroids \((M_i)_{i\in I}\), there exists a projective plane \(\Pi\) containing each \(M_i\) such that any choice of orientations \(\chi_i\) of \(M_i\), \(i\in I\), can be extended to an orientation \(\chi\) of \(\Pi\). If the matroids \(M_i\) considered are finite, then \(\chi\) can be chosen to induce even an archimedean ordering on \(\Pi\).” The connection between ordered planes and oriented matroids is made via A. W. M. Dress’ and W. Wenzel’s theory of Tutte groups [Adv. Math. 77, 1-36 (1989; Zbl 0684.05013)].

MSC:
52C40 Oriented matroids in discrete geometry
51E15 Finite affine and projective planes (geometric aspects)
51G05 Ordered geometries (ordered incidence structures, etc.)
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