Max Semigroup on Toset forms Semilattice
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Theorem
Let $\struct {S, \preceq}$ be a totally ordered set.
Then the max semigroup $\struct {S, \max}$ is a semilattice.
Proof
The Max Semigroup is Commutative and idempotent.
Hence the result, by definition of a semilattice.
$\blacksquare$