Contraction, loops and flats.

Question

This idea is being used a lot, but I cannot justify why it is correct:

If M is a matroid and $T$ a subset of $E(M).$ Then $M/T$ has no loops iff $T$ is a flat of $M.$

I know how to proof that in a loopless matroid every component is flat. But I still do not know how to justify this idea.

Can someone explain to me why this idea is correct please?

Answer 1

Let's recall some definitions. $T\subseteq E(M)$ is a flat if $r(T+e)>r(T)$ for any $e\in E(M)-T$, i.e., $\operatorname{cl}(T) = T$. A loop in a matroid is an element of zero rank.

Suppose $T$ is a flat of $M$. Then for any $e\in E(M)-T$

$$ r_{M/T}(e) = r_M(T+e) -r_M(T) > 0, $$ i.e., $e$ is not a loop. The converse follows by following the argument backwards. Basically, this all follows from the definition of contraction.