Idempotent Elements form Submonoid of Commutative Monoid

Theorem

Let $\struct {S, \circ}$ be a commutative monoid.

Let $e \in S$ be the identity element of $\struct {S, \circ}$.

Let $I$ be the set of all elements of $S$ that are idempotent under $\circ$.

That is:

$I = \set {x \in S: x \circ x = x}$

Then $\struct {I, \circ}$ is a submonoid of $\struct {S, \circ}$ with identity $e$.

Proof

By Idempotent Elements form Subsemigroup of Commutative Semigroup, $\struct {I, \circ}$ is a subsemigroup of $\struct {S, \circ}$.

By Identity Element is Idempotent:

$e \in I$

By Identity of Algebraic Structure is Preserved in Substructure, $e$ is an identity of $\struct {I, \circ}$.

Since $\struct {T, \circ}$ is a semigroup and has an identity, $\struct {T, \circ}$ is a monoid.

Since $T \subseteq S$, $\struct {T, \circ}$ is a submonoid of $\struct {S, \circ}$ with identity $e$.

$\blacksquare$