Positive Real Numbers under Max Operation form Monoid

Theorem

Let $\R_{\ge 0}$ be the set of positive (that is, non-negative) real numbers.

Let $\max: \R_{\ge 0}^2 \to \R_{\ge 0}$ be the max operation on $\R_{\ge 0}$.

Then $\struct {\R_{\ge 0}, \max}$ is a monoid whose identity is $0$.

Proof

From Real Numbers are Totally Ordered, $\R$ is a totally ordered set.

From Max Operation on Toset forms Semigroup, $\struct {\R_{\ge 0}, \max}$ is a semigroup.

By definition of $\R_{\ge 0}$:

$\forall x \in \R_{\ge 0}: 0 \le x$

Thus by definition of the max operation:

$\forall x \in \R_{\ge 0}: \map \max {0, x} = x = \map \max {x, 0}$

So $0$ is the identity of $\struct {\R_{\ge 0}, \max}$ by definition.

The result follows by definition of monoid.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids