Let $M$ be a matroid on a finite set $E$ and $A\subset E$. I want to define a span of $A$, but not as a subset of $M$ (which could be defined as a union of all $B\supset A$ satisfying ${\rm rank}\, B={\rm rank}\, A$): I would like to consider elements not lying in $M$. This can be defined as a lattice of classes of extensions of $M$ to matroids on larger finite sets $E\sqcup E_1$ satisfying ${\rm rank}\, A\sqcup E_1={\rm rank}\, A$. If $M$ is representable over a field, this is like a genuine span, at least in the sense of matroid structures on finite subsets. Are there references to such a notion? Is there another, more direct definition?

Let $M$ be a matroid on a finite set $E$ and $A\subset E$. I want to define a span of $A$, but not as a subset of $M$ (which could be defined as a union of all $B\supset A$ satisfying ${\rm rank}\, B={\rm rank}\, A$): I would like to consider elements not lying in $M$. This can be defined as a lattice of classes of extensions of $M$ to matroids on larger finite sets $E\sqcup E_1$ satisfying ${\rm rank}\, A\sqcup E_1={\rm rank}\, A$. If $M$ is representable over a field (and all extensions which we consider also are), this is like a genuine span, at least in the sense of matroid structures on finite subsets. Are there references to such a notion? Is there another, more direct definition?