Oriented Matroid

An oriented matroid can be represented (and defined) in several, equivalent ways. The basic axiom systems include vector (or covector) axioms, circuit (or cocircuit) axioms, and chirotopes.

We define here an oriented matroid using the covector axioms: An oriented matroid is a pair (EF) of a finite set E and a set F of sign vectors (called covectors) on E for which the following covector axioms (F0) to (F3) are valid:
(F0) 0 = 00..0 in F.  
(F1) If X in F then -X in F. (symmetry)
(F2) If X,Y in F then XoY in F,
where (XoY)e = Xe if Xe not 0 and (XoY)e = Ye otherwise.
(F3) For all X,Y in F and e in D(X,Y) = {g in E | Xg = -Yg not 0}
there exists X in F such that Ze = 0 and Zf = (XoY)f for all f in E \ D(X,Y).
(covector elimination)

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