Min Operation on Toset is Semigroup
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Theorem
Let $\struct {S, \preceq}$ be a totally ordered set.
Let $\map \min {x, y}$ denote the min operation on $x, y \in S$.
Then $\struct {S, \min}$ is a semigroup.
Proof
By the definition of the min operation, either:
- $\map \min {x, y}= x$
or
- $\map \min {x, y}= y$
So $\min$ is closed on $S$.
From Min Operation is Associative:
- $\forall x, y, z \in S: \map \min {x, \map \min {y, z} } = \map \min {\map \min {x, y}, z}$
Hence the result, by definition of semigroup.
$\blacksquare$