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: finite monoids, free semigroups, free commutative monoids, 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 left- nor 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 semigroupsto 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 automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as a cornerstone of our construction.

Divisibility monoids are a natural algebraic generalisation of Mazurkiewicz trace monoids, namely monoids in which the distributivity of the underlying lattices is kept as a hypothesis, but the relations between the generators are not necessarily supposed to be commutations. Garside monoids are a natural algebraic generalisation of spherical Artin monoids, namely monoids in which the existence of common multiples is kept as a hypothesis, but the relations between the generators are not necessarily supposed to be of Coxeter type. Here, we show that the quasi-centre of these monoids can be studied and described in a similar way, and then we exhibit the intersection between the two classes of monoids.