# Max Operation on Toset is Semigroup

## Theorem

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

Let $\map \max {x, y}$ denote the max operation on $x, y \in S$.

Then $\struct{S, \max}$ is a semigroup.

## Proof

By the definition of the max operation, either:

$\map \max {x, y}= x$

or

$\map \max {x, y}= y$

So $\max$ is closed on $S$.

$\forall x, y, z \in S: \map \max {x, \map \max {y, z}} = \map \max {\map \max {x, y}, z}$

Hence the result, by definition of semigroup.

$\blacksquare$