Integers under Multiplication form Semigroup
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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: $(1)$