Max Operation on Toset forms Semigroup
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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 $S$.
From Max Operation is Associative:
- $\forall x, y, z \in S: \map \max {x, \map \max {y, z}} = \map \max {\map \max {x, y}, z}$
Hence the result, by definition of semigroup.
$\blacksquare$