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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 (E, F) 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 X°Y in F, where (X°Y)_{e} = X_{e} if X_{e} not 0 and (X°Y)_{e} = Y_{e} otherwise. |
(composition) |
(F3) | For all X,Y in F and
e in D(X,Y) =
{e in E | X_{e} = -Y_{e} not 0} there exists Z in F such that Z_{e} = 0 and Z_{f} = (X°Y)_{f} for all f in E \ D(X,Y). |
(covector elimination) |