Directed Acyclic Graph Ancestry Partial Order

Why

Ancestry in a directed acyclic graph gives a partial order over the vertices.

Suppose $(V, E)$ is a directed acyclic graph. Define the relation $\preceq$ by

\[ s \preceq t \text{ if } s \text{ is an ancestor of } t \quad \text{ for all } s, t \in V \]

Then $\preceq$ is a partial order. We call it the ancestry partial order.