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Questions related to the field of Combinatorics called Matroid Theory. Relevant topics include matroids in Combinatorial Optimization, Lattice Theory, Algebraic Geometry, Polyhedral Theory, Rigidity, and Algorithms. For questions about Oriented Matroids, the oriented-matroids tag may be used.
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matroids axioms and independence system
The substantial question 2 being already answered, I will address the less substantial 1.
Even if not substantial, the formalization of 1 might not be immediate (and at the end all the question might …