# 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 $\left({S, \max}\right)$ denote the algebraic structure formed from $S$ and the max operation.

Then $\left({S, \max}\right)$ 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): $\S 8$: Example $8.4$