Integers under Addition form Semigroup
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Theorem
The set of integers under addition $\struct {\Z, +}$ forms a semigroup.
Proof
Closure
$\Box$
Associativity
Integer Addition is Associative.
$\Box$
The criteria for $\struct {\Z, +}$ to be a semigroup are seen to be satisfied.
$\blacksquare$
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 $4$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 29$. Semigroups: definition and examples