Divisor Relation on Positive Integers is Partial Ordering

Theorem
The divisor relation is a partial ordering of $\Z_{>0}$.

Proof
Checking in turn each of the criteria for an ordering:

Divisor Relation is Transitive

 * $\forall x, y, z \in \Z: x \divides y \land y \divides z \implies x \divides z$:

Divisor Ordering is Partial
Let $a = 2$ and $b = 3$.

Then neither $a \divides b$ nor $b \divides a$.

Thus, while the divisor relation is an ordering, it is specifically a partial ordering