Automatic semigroups vs automaton semigroups

We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup, namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication. The framework embraces many of the well-known classes of (automatic) semigroups: free semigroups, free commutative semigroups, trace or divisibility monoids, braid or Artin-Tits or Krammer or Garside monoids, Baumslag-Solitar semigroups, etc. Like plactic monoids or Chinese monoids, some neither leftnor right-cancellative automatic semigroups are also investigated, as well as some residually finite variations of the bicyclic monoid. It provides what appears to be the first known connection from a class of automatic semigroups to a class of automaton semigroups. It is worthwhile noting that, in all these cases, "being an automatic semigroup" and "being an automaton semigroup" become dual properties in a very automatatheoretical sense. 1998 ACM Subject Classification F.1.1 Models of Computation, F.4.3 Formal Languages.