Integers under Multiplication form Semigroup

Theorem

The set of integers under multiplication $\struct {\Z, \times}$ is a semigroup.

Proof

Semigroup Axiom $\text S 0$: Closure

Integer Multiplication is Closed, fulfilling Semigroup Axiom $\text S 0$: Closure.

$\Box$

Semigroup Axiom $\text S 1$: Associativity

Integer Multiplication is Associative, fulfilling Semigroup Axiom $\text S 1$: Associativity.

$\Box$

Hence the semigroup axioms are seen to be fulfilled.

Thus $\struct {\Z, \times}$ is a semigroup.

$\blacksquare$

Sources

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$: Definition and examples of semigroups: Example $5$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 29$. Semigroups: definition and examples: $(1)$