Home | Catalog of Oriented Matroids | Catalog of Point Configurations | Catalog of Hyperplane Arrangements | Glossary |
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
$$XoY in F, where $($XoY)_{e} = X_{e} if $$X_{e} not 0 and $($XoY)_{e} = Y_{e}$otherwise.$ |
(composition) |
(F3) |
For all $$X,Y in F
and $$e in D(X,Y) = {g in
E | X_{g} = -Y_{g} not 0} there exists $$X in F such that $$Z_{e} = 0 and $$Z_{f} = (XoY)_{f} for all $$f in E \ D(X,Y). |
(covector elimination) |
Maintained by Dr. Lukas Finschi | Terms and Conditions | Privacy Policy |
---|