## Oriented matroids in terms of order functions.(English)Zbl 0882.05041

Summary: An oriented matroid of rank $$r$$ is described as a pair $$(E,\chi)$$ 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 $$a,b\in E\backslash H$$, $$\omega$$ is shown to be harmonic and strict. The presented theorem gives a geometric characterization of oriented matroids in terms of order functions.

### MSC:

 05B35 Combinatorial aspects of matroids and geometric lattices

### Keywords:

oriented matroid; chirotope; order function; hyperplane
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