On the Reduciblity of a Certain Type of Rank 3 Uniform Oriented Matroid by a Point

Abstract

For a positive integer $n\geq 3$, the sides and diagonals of a convex $n$-gon divide the interior of the convex $n$-gon into finitely (polynomial in $n$) many regions bounded by them. In this article, we associate to every region a unique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as $2$-standard consecutive cycle) by studying point arrangements in the plane. Then we find that there are more (exponential in $n$) number of such cycles leading to the conclusion that not every region labelled by a cycle appears in every convex $n$-gon. In fact most of them do not occur in any given single convex $n$-gon. Later in the main theorem of this article we characterize combinatorially those cycles (defined as definite cycles) whose corresponding regions occur in every convex $n$-gon and those cycles (defined as indefinite cycles) whose corresponding regions do not occur in every convex $n$-gon. As a consequence we characterize those one point extensions of a uniform rank $3$ convex oriented matroid for which the one point extension is reducible by the, one point, when it lies inside the convex hull.