The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity

We extend the notion of activity for automaton semigroups and monoids introduced by Bartholdi, Godin, Klimann and Picantin to a more general setting. Their activity notion was already a generalization of Sidki's activity hierarchy for automaton groups. We show that the language of $\omega$-words with infinite orbits is effectively a deterministic B\"uchi language for automata with bounded extended activity, which yields decidability of the finiteness problem for complete automaton semigroups and monoids of bounded activity (solving an open problem by Bartholdi, Godin, Klimann and Picantin). In fact, we obtain a stronger result also covering finitely generated subsemigroups.