I've read some articles, and see free representations and some aspects detailed on the relations between Matroids and Modules. Would like to receive advices about readings and more materials about this subject.
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...
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