Equivalent Conditions for Element is Loop

Theorem

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

Let $\sigma$ denote the closure operator on $M$.

Let $\rho$ denote the rank function of $M$.

Let $\mathscr B$ denote the set of all bases of $M$.

Let $x \in S$.

The following statements are equivalent:

$(1)\quad x$ is a loop

$(2)\quad x \in \map \sigma \O$

$(3)\quad \map \rho {\set x} = 0$

$(4)\quad \set x$ is a circuit

$(5)\quad x$ is not an element of any $B \in \mathscr B$

Proof

Condition $(1)$ iff Condition $(2)$

Follows immediately from Element is Loop iff Member of Closure of Empty Set.

$\Box$

Condition $(1)$ iff Condition $(3)$

Follows immediately from Element is Loop iff Rank is Zero.

$\Box$

Condition $(1)$ iff Condition $(4)$

Follows immediately from Element is Loop iff Singleton is Circuit.

$\Box$

Condition $(1)$ iff Condition $(5)$

Follows immediately from the contrapositive statement of Element is Member of Base iff Not Loop.

$\blacksquare$