Proposition: Strongly connected quotient

For an arbitrary directed graph, define

Reachability induces a partial order on $E/\sim$.

Proof. Reflexivity and transitivity make $\sim$ an equivalence relation. If $p\sim p^{\prime }$, $q\sim q^{\prime }$, and $p\le q$, then $p^{\prime }\le p\le q\le q^{\prime }$. Thus the quotient relation is well-defined. Mutual quotient reachability means equality of equivalence classes.