  We introduce the category of dependency automata. A dependency automaton
consists of two nondeterministic finite automata, with a relation between their
states satisfying conditions. This category is equivalent to deterministic
finite automata interpreted in join-semilattices i.e. JSL-dfas. The canonical
dependency automaton accepting $L$ amounts to the state-minimal dfas for $L$
and $rev(L)$ connected by the `dependency relation'.
  We describe many canonical JSL-dfas as dependency automata and also
explain/extend Brzozowski's algorithm. Call an nfa `subatomic' if its
individual states accept a language in the closure of $\{L\}$ under left/right
quotients and set-theoretic boolean operations. We prove an nfa $N$ is
subatomic iff $rsc(rev(N))$'s transition monoid is syntactic.
