What is this notation $( \mathbb{N},|)$

Question

My book has this notation for examples of a poset $( \mathbb{N},|)$ and $(\mathbb{Z^*},|)$ and I need to verify it if it is a partially ordered set. Can someone tell me how to read this notation?

Answer 1

In my opinion, $\mid$ represents the divisibility relation here, i.e. consider $\mid$ as

$$\mid\;\; =\{(a,b)\in\mathbb N^2 : a\mid b\;(a\text{ divides }b)\}$$

and similar for $\mathbb Z^*$.

Answer 2

I believe that it is to denote the division operation? That is the most probable reason they're notated that way, much like in group theory and abstract algebra