Element is Loop iff Member of Closure of Empty Set

Theorem

Let $M = \struct{S, \mathscr I}$ be a matroid.

Let $x \in S$.

Then:

$x$ is a loop if and only if $x \in \map \sigma \O$

where $\map \sigma \O$ denotes the closure of the empty set.

Proof

From Element is Loop iff Rank is Zero:

$x$ is a loop if and only if $\map \rho {\set x} = 0$

where $\rho$ is the rank function of $M$.

Now:

\(\ds \)

\(\)

\(\ds x \in \map \sigma \O\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds x \sim \O\)

Definition of Closure Operator

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \map \rho {\set x} = \map \rho \O\)

Definition of Depends Relation

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \map \rho {\set x} = 0\)

Rank of Empty Set is Zero

The result follows.

$\blacksquare$

Sources

• 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 4.$ Loops and parallel elements