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

Jaritz, Renate

Beitr. Algebra Geom. 38, No. 2, 249-260 (1997).

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