Integer Multiples under Multiplication form Semigroup

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Theorem

Let $n \Z$ be the set of integer multiples of $n$.

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

If $\size n > 1$ then $\struct {n \Z, \times}$ has no identity.


Proof

Closure

Let $p, q \in n \Z$.

Then for some $p', q' \in \Z$:

$p = n p'$
$q = n q'$

Hence:

$p q = \paren {n p'} \paren {n q'}$

By the commutativity and associativity of integer multiplication:

$p q = n \paren {n \paren {p' q'} }$

Hence $p q \in n \Z$ and $n \Z$ is closed under $\times$.

$\Box$


Assocativity

By definition $n \Z \subseteq \Z$.

Hence $\times$ is associative on $n \Z$ as a direct results of Restriction of Associative Operation is Associative.

$\Box$


Identity Element

If $\size n = 1$ then $\paren {n \Z, \times}$ is a monoid from Integers under Multiplication form Countably Infinite Commutative Monoid.

If $n \ne 1$ then $\left({n \Z, \times}\right)$ is a semigroup but not a monoid as it is without an identity as $1 \notin n \Z$.

$\blacksquare$


Sources

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