Oriented matroid

Remarks

The supports of the signed circuits are the circuits of a matroid on $E$, the underlying matroid. Axiom 4 is the signed form of circuit elimination. There are equivalent axiom systems in terms of covectors, of vectors and of chirotopes, which are signed versions of the bases.

Example: let $v_1, \dots, v_n$ be vectors in $\R^r$ and $E = \set{1, \dots, n}$. For each linear dependence $\sum_i \lambda_i v_i = 0$ whose support $\set{i : \lambda_i \ne 0}$ is minimal and nonempty, put $X^+ = \set{i : \lambda_i > 0}$ and $X^- = \set{i : \lambda_i < 0}$. These signed sets satisfy the axioms; such oriented matroids are called realizable. For $v_1 = e_1$, $v_2 = e_2$, $v_3 = e_1 + e_2$ in $\R^2$, the only minimal dependence up to scaling is $v_1 + v_2 - v_3 = 0$, so $\mathcal{C} = \set{(\set{1,2}, \set{3}), (\set{3}, \set{1,2})}$.

Non-example: on $E = \set{1, 2}$, the set $\mathcal{C} = \set{(\set{1}, \set{2}), (\set{1,2}, \emptyset), (\set{2}, \set{1}), (\emptyset, \set{1,2})}$ fails axiom 3, since all four sets have the same support but $(\set{1}, \set{2}) \ne \pm (\set{1,2}, \emptyset)$.