Natural Numbers under Multiplication form Semigroup

Theorem

Let $\N$ be the set of natural numbers.

Let $\times$ denote the operation of multiplication on $\N$.

The structure $\struct {\N, \times}$ forms a semigroup.

Proof

Semigroup Axiom $\text S 0$: Closure

We have that Natural Number Multiplication is Closed.

That is, $\struct {\N, \times}$ is closed.

$\Box$

Semigroup Axiom $\text S 1$: Associativity

We have that Natural Number Multiplication is Associative.

$\Box$

Thus the criteria are fulfilled for $\struct {\N, \times}$ to form a semigroup.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 29$. Semigroups: definition and examples: $(1)$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): semigroup
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): semigroup