Extended Real Numbers under Multiplication form Monoid

Theorem

Denote with $\overline \R$ the extended real numbers.

Denote with $\cdot_{\overline \R}$ the extended real multiplication.

The algebraic structure $\struct {\overline \R, \cdot_{\overline \R} }$ is a monoid.

Checking the axioms for a monoid in turn:

Immediate as $\cdot_{\overline \R}: \overline \R \times \overline \R \to \overline \R$ is a mapping.

$\Box$

$\Box$

For all $x \in \R$, it holds that $1 \cdot_{\overline \R} x = x \cdot_{\overline \R} 1 = x$ by definition of $\cdot_{\overline \R}$ on $\R$.

Furthermore, by definition, $1 \cdot_{\overline \R} \paren {+\infty} = \paren {+\infty} \cdot_{\overline \R} 1 = \paren {+\infty}$.

Lastly $1 \cdot_{\overline \R} \paren {-\infty} = \paren {-\infty} \cdot_{\overline \R} 1 = \paren {-\infty}$.

That is, $1 \in \overline \R$ is an identity for $\cdot_{\overline \R}$.

$\Box$

Hence, satisfying all the axioms, $\struct {\overline \R, \cdot_{\overline \R} }$ is a monoid.

$\blacksquare$