I'm reading Some applications of the method of forcing by Todorchevich and Farah and one of the exercises seems to be wrong to me.
Definitions
We fix some partially ordered set $(P,\preceq)$.
A filter on the poset is a set $G\subseteq P$ such that
- whenever $p\in G$ and $p\preceq q$ then $q\in G$, and
- for all $p,q\in G$, there is $r\in G$ such that $r\preceq q$ and $r\preceq p$.
The set $D\subseteq P$ is dense if for all $p\in P$, there is $q\in D$ such that $q\preceq p$.
An atom is an element $p\in P$ such that the set $\{q\in P:q\preceq p\}$ is linearly ordered by $\preceq$.
Exercise
There is a filter $G$ that intersects every dense set (a generic filter) if and only if there is an atom in $P$.
Counterexample
Let $P=\mathbb{Z}$ and let $p\preceq q$ iff $|p|>|q|$ or $p=q$. Then $(P,\preceq)$ has no atom, but $G=P$ is a filter that intersects every subset and hence every dense subset.
Is there something wrong with my counterexample? If not, is there some version of the exercise that is related to forcing and is correct?
Since there was some speculation what is actually in the text, here is the relevant part of the first page: