# Finding dual lattice of $(\Bbb N_0,\text{divisibility})$ and showing its isomorphism

I'm trying to understand the concept of dual lattices by solving a problem:

Prove that the dual lattice of $(\Bbb N_0,\text{divisibility})$ is isomorphic with the lattice $\text{sub}(\Bbb Z)$ where $(\Bbb Z,+)$ is the group of all integers.

I'm not really sure how the dual lattice is found for lattices like these (in order theory). I only know the dual lattice definition (in group theory):

The dual of a lattice $\wedge$ is the set $\hat{\wedge}$ of all vectors $\mathbf{x}\in\text{span}(\wedge)$ such that $\langle \mathbf{x,y} \rangle$ is an integer for $\mathbf{y}\in\wedge$

which doesn't seem to be applicable here.

Any idea how to find the dual lattice and then show the isomorphism?

We obtain the dual lattice if we simply reverse its partial order, or equivalently, if we exchange the lattice operations $\land$ and $\lor$.
For the statement, you basically have to prove that all subgroups of $\Bbb Z$ are of the form $n\Bbb Z$ for some $n\in\Bbb Z$.