Element is Loop iff Singleton is Circuit

Theorem

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

Let $x \in S$.

Then:

$x$ is a loop if and only if $\set x$ is a circuit

Proof

Necessary Condition

Let $x$ be a loop.

By definition of a loop:

$\set x$ is a dependent subset of $S$

Let $A \subseteq \set x$ be a dependent subset.

From Power Set of Singleton:

$\powerset {\set x} = \set{\O, \set x}$

By matroid axiom $(\text I 1)$:

$\O$ is an independent subset

Then:

$A = \set x$

It follows that:

$\set x$ is a minimal dependent subset of $S$.

Then $\set x$ be a circuit by definition.

$\Box$

Sufficient Condition

Let $\set x$ be a circuit.

By definition of a circuit:

$\set x$ is a minimal dependent subset of $S$.

In particular, $\set x$ is a dependent subset of $S$.

Then $\set x$ is a loop by definition.

$\blacksquare$

Sources

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