Integers under Addition form Semigroup

Theorem

The set of integers under addition $\struct {\Z, +}$ forms a semigroup.


Proof

Closure

Integer Addition is Closed.

$\Box$


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
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