Definition:Matroid Induced by Linear Independence/Abelian Group

Definition
Let $\struct{G, +}$ be a torsion-free Abelian group.

Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$.

Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $G$.

Let $\mathscr I$ be the set of linearly independent subsets of $S$.

Then $\struct{S, \mathscr I}$ is called the matroid induced by linear independence in $G$ on $S$.

Also see

 * Leigh.Samphier/Sandbox/Matroid Induced by Torsion-Free Group is Matroid