[I'm aware that this question is opinion-based - because it's about "interesting" properties - but I dare to ask it nevertheless. You can replace "interesting" by "important" if you want to – which doesn't make the question less opinion-based.]
I start from the observation that there is a strong formal analogy between the "interesting" concepts of being prime and of being co-prime:
$\alpha$ is prime iff
$$(\forall xy)\ \alpha|x \vee \alpha|y \leftrightarrow \alpha|xy$$
$\alpha, \beta$ are co-prime iff
$$(\forall x)\ \alpha|x \wedge \beta|x \leftrightarrow \alpha\beta|x$$
and that there is presumably an "interesting" relation between the two concepts.
Now I wonder if this might be a general scheme for arbitrary relations $\phi(x,y)$:
If a property P defined like this
$\alpha$ is P iff
$$(\forall xy)\ \phi(\alpha,x) \vee \phi(\alpha,y) \leftrightarrow \phi(\alpha,xy)$$
is "interesting", then also the relation co-P defined like this
$\alpha, \beta$ are co-P iff
$$(\forall x)\ \phi(\alpha,x) \wedge \phi(\beta,x) \leftrightarrow \phi(\alpha\beta,x) $$
is "interesting", and both relate in an "interesting" way.
If not for every "interesting" property P: for which properties? (Examples or a criterion would be welcome.)
But maybe P = prime with $\phi(x,y) = (\exists z)\ xz = y$ is the only such "interesting" property.
Note that in groups and in fields this relation is not interesting because it holds for all $x,y$ (in fields: with $x\neq 0$) and no $x,y$ are co-prime.
