  The article continues our study of the genus of a regular language $L$,
defined as the minimal genus among all genera of all finite deterministic
automata recognizing $L$. Here we define and study two closely related tools on
a directed graph: directed emulators and automatic relations. A directed
emulator morphism essentially encapsulates at the graph-theoretic level an
epimorphism onto the minimal deterministic automaton. An automatic relation is
the graph-theoretic version of the Myhill-Nerode relation. We show that an
automatic relation determines a directed emulator morphism and respectively, a
directed emulator morphism determines an automatic relation up to isomorphism.
Consider the set $S$ of all directed emulators of the underlying directed graph
of the minimal deterministic automaton for $L$. We prove that the genus of $L$
is $\underset{G \in S}{\min}\ g(G)$. We also consider the more restrictive
notion of directed cover and prove that the genus of $L$ is reached in the
class of directed covers of the underlying directed graph of the minimal
deterministic automaton for $L$. This stands in sharp contrast to undirected
emulators and undirected covers which we also consider. Finally we prove that
if the problem of determining the minimal genus of a directed emulator of a
directed graph has a solution then the problem of determining the minimal genus
of an undirected emulator of an undirected graph has a solution.
