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Pré-Publication, Document De Travail Année : 2017

Automatic semigroups vs automaton semigroups

Résumé

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 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 automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as a cornerstone of our construction.
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Dates et versions

hal-01373972 , version 1 (29-09-2016)
hal-01373972 , version 2 (06-12-2016)
hal-01373972 , version 3 (16-01-2017)

Identifiants

Citer

Matthieu Picantin. Automatic semigroups vs automaton semigroups. 2017. ⟨hal-01373972v3⟩
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