Min Semigroup on Toset forms Semilattice

Theorem

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

Then the min semigroup $\struct {S, \min}$ is a semilattice.

Proof

We have:

Min Semigroup is Commutative

Min Semigroup is Idempotent.

Hence the result, by definition of a semilattice.

$\blacksquare$

Also see

• Max Semigroup on Toset forms Semilattice