Divisor Relation induces Lattice

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Theorem

Let $\left({\Z_{> 0}, \backslash}\right)$ be the ordered set comprising:

The set of positive integers $\Z_{> 0}$
The divisor relation $\backslash$ defined as:
$a \mathrel \backslash b := \exists k \in \Z_{> 0}: b = ka$


Then $\left({\Z_{> 0}, \backslash}\right)$ is a lattice.


Proof

It follows from Divisor Relation on Positive Integers is Partial Ordering that $\left({\Z_{>0}, \backslash}\right)$ is indeed an ordered set.


Let $a, b \in \Z_{>0}$.

Let $d = \gcd \left\{{a, b}\right\}$ be the greatest common divisor of $a$ and $b$.

By definition, $d$ is the infimum of $\left\{{a, b}\right\}$.


Similarly, let $m = \operatorname{lcm} \left\{{a, b}\right\}$ be the lowest common multiple of $a$ and $b$.

By definition, $m$ is the supremum of $\left\{{a, b}\right\}$.


Hence, as $\left\{{a, b}\right\}$ has both an infimum and a supremum in $\Z_{>0}$, it follows that $\left({\Z_{> 0}, \backslash}\right)$ is a lattice.

$\blacksquare$


Sources