Oriented Matroids in Terms of Order Functions

Abstract: An oriented matroid of rank $r$ is described as a pair ${\cal M} = ( E, \chi )$ where $E$ is a nonempty set and $\chi$ is a chirotope of rank $r$ on $E$. Relating the chirotope to an order function $\omega : (H,a,b) \mapsto (H|ab)$ according to the rule $(H|ab) = \chi(P,a)\cdot\chi(P,b)$ where $H$ is a hyperplane, $P$ is a base of $H$ and $a$ and $b$ are points with $a,b \in E\setminus H$, $\omega$ is shown to be harmonic and strict. The presented theorem gives a geometric characterization of oriented matroids in terms of order functions.

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