Natural Numbers under Multiplication form Subsemigroup of Integers

Theorem

Let $\struct {\N, \times}$ denote the set of natural numbers under multiplication.

Let $\struct {\Z, \times}$ denote the set of integers under multiplication.

Then $\struct {\N, \times}$ is a subsemigroup of $\struct {\Z, \times}$.

Proof

We have from Natural Numbers under Multiplication form Semigroup that $\struct {\N, \times}$ forms a semigroup.

We have from Integers under Multiplication form Semigroup that $\struct {\Z, \times}$ forms a semigroup.

From Natural Numbers are Non-Negative Integers, we have that $\N \subseteq \Z$.

From the definition of integer multiplication it follows that $\times: \Z \to \Z$ is an extension of $\times: \N \to \N$.

Hence the result.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 32$ Identity element and inverses