When are dual Cayley automaton semigroups finite?
  In this note we prove that, for a finite semigroup $S$, the dual   Cayley automaton semigroup $\mathbf{C^{\ast}}(S)$ is finite if and only if $S$ is $\mathcal{H}$-trivial and has no non-trivial right zero subsemigroups. 
Victor Maltcev