Oriented rank three matroids and projective planes
Kalhoff shows that classical work about order functions and half-orderings of partial planes by \textit{J. Joussen} [Result. Math. 4, 55-74 (1981; Zbl 0466.51003)] together with the completion theorem of \textit{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 \textit{J. E. Goodman, R. Pollack, R. Wenger} and \textit{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 \textit{A. W. M. Dress}' and \textit{W. Wenzel}'s theory of Tutte groups [Adv. Math. 77, 1-36 (1989; Zbl 0684.05013)].
scientific article; zbMATH DE number 1436082
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