Subset of Natural Numbers under Max Operation is Monoid

From ProofWiki
Jump to navigation Jump to search

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