Closure of a matroid (${\mathrm{cl}}_M(A)=\{x:r(A\cup \{x\})=r(A)\}$)

The closure of a matroid adds all elements that fail to increase its rank of a matroid. It abstracts the linear span in a linear matroid. It is extensive, preserves inclusion, preserves the rank of a matroid, and is an idempotent function:Proof. Extensivity follows from $A+x=A$ when $x\in A$. If $A\subseteq B$ and $x\notin B$, submodularity of matroid rank gives $r(B+x)-r(B)\le r(A+x)-r(A)$. Thus an element in ${\mathrm{cl}}_M(A)$ also lies in ${\mathrm{cl}}_M(B)$; the case $x\in B$ is automatic. Add the finitely many elements of ${\mathrm{cl}}_M(A)\setminus A$ one at a time. Inclusion preservation makes each addition have zero marginal increment of the rank of a matroid, proving rank preservation.

For completeness, nested subsets $A\subseteq B$ of equal rank of a matroid have the same closure of a matroid. If $x\notin B$, apply submodularity of matroid rank to $B$ and $A+x$ to obtain $r(B+x)\le r(A+x)$; monotonicity gives the reverse bound. The defining tests for membership in their closures are therefore equal. If $x\in B$, $r(A)\le r(A+x)\le r(B)=r(A)$, so both closures contain $x$. Apply this equal-rank result to $A\subseteq {\mathrm{cl}}_M(A)$ to prove idempotence. In particular, every maximal matroid independent set of $B$ has the same closure of a matroid as $B$.