Visualizing directed preorders I am trying to understand directed preorders, a.k.a. directed sets. Are they analogous to connected DAGs?

Not quite. A DAG cannot have edges $x\to y$ and $y\to x$, while a directed preorder can have $x\leq y$ and $y\leq x$.
In general, you can visualize a preorder as the reachability relation on the vertices of a directed graph. Then a directed preorder corresponds to a graph where any two vertices are reachable from some third vertex.
