Subset of Natural Numbers under Max Operation is Monoid
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Theorem
Let $S \subseteq \N$ be a subset of the natural numbers $\N$.
Let $\struct {S, \max}$ denote the algebraic structure formed from $S$ and the max operation.
Then $\struct {S, \max}$ is a monoid.
Its identity element is the smallest element of $S$.
Proof
By the Well-Ordering Principle, $\N$ is a well-ordered set.
By definition, every subset of a well-ordered set is also well-ordered.
Thus $S$ is a well-ordered set.
The result follows from Max Operation on Woset is Monoid.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.4$