Max Operation on Toset is 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.

$\blacksquare$


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