Max Semigroup is Idempotent

Theorem

Let $\struct {S, \preceq}$ be a totally ordered set.

Then the semigroup $\struct {S, \max}$ is an idempotent semigroup.

Proof

The fact that $\struct {S, \max}$ is a semigroup is demonstrated in Max Operation on Toset forms Semigroup.

Then the max operation is idempotent:

$\forall x \in S: \max \set {x, x} = x$

The result follows by the definition of idempotent semigroup.

$\blacksquare$

Also see

• Min Semigroup is Idempotent