just define a matroid.

There are a lot of different and analogous definitions. You can define a Matroid as a closure space satisfying: (i) if y ∈ cl(A ∪ x)\cl(A) for some x ∈ M, then x ∈ cl(A ∪ y) (the exchange

property), and (ii) if x ∈ cl(A), then x ∈ cl(A]) for some finite A' ⊆ A (finitary). This one comes from: https://www.researchgate.net/publication/220369923_Matroids_from_Modules   And the basic is a similiar given in: https://www.researchgate.net/publication/222424536_Representations_of_matroids_and_free_resolutions_for_multigraded_modules A matroid M on a finite set S is given by a nonempty collection I of

subsets of S that satisfies the following two properties:

(i) If X ⊂ Y and Y ∈ I then X ∈ I;

(ii) If X, Y ∈ I, and |Y | = |X|+1 then there is y ∈ Y \ X such that X∪{y} ∈ I.   You can find these articled on ArXiv or some database.

Article Matroids from Modules

Article Representations of matroids and free resolutions for multigr...