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Automatic semigroups vs automaton semigroups

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Matthieu Picantin (IRIF)

Group Theory · Submitted 29 Sep 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It provides what appears to be the first known connection from a class of automatic semigroupsto a class of automaton semigroups.

how it is stated · hedged possibility 0.72 read the record →

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arXiv ID1609.09364 (version 5)
Pith IDRK5W26ZJ, Pith's permanent record of this paper and its review
CitationMatthieu Picantin (IRIF). "Automatic semigroups vs automaton semigroups." arXiv:1609.09364 (2016). https://pith.science/paper/RK5W26ZJ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,777 input tokens.

What the abstract itself states, as probabilities: claims first 0.86 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 0.80 0.94 0.52 asserted 1.00
s1 It provides what appears to be the first known connection from a class of automatic semigroupsto a class of automaton semigroups. result 0.50 0.85 0.51 hedged possibility 0.72
s2 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. interpretation 0.98 0.68 0.16 asserted 0.99
s3 Quadratic rewriting systems and associated tilings appear as a cornerstone of our construction. method 0.97 0.76 0.06 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.83 · evidence from proof or derivation 0.99.

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The paper
Original abstract

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.

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Cites as background 2 papers
  • we just recall that semigroups could appear much more difficult to handle, especially when it comes to automaticity (see [ref] ) or selfsimilarity (see [ref] ).

    Automaton semigroups: new construction results and examples of non-automaton semigroups · From an automatic structure to a selfsimilar structure · 0.99

  • The latter happens to be the common smallest nontrivial member of the family of Baumslag-Solitar monoids (see [ref] for instance), namely $^1_+(1,0)$, and of a wide family of right-cancellative semigroups, that we readily call Artin-Krammer monoids and that have been introduced and studied in [ref] (see also [ref] ), namely $^1_+(Γ)$ associated with the Coxeter-like matrix $Γ=$.7$[ 1 1; 2 1 ]$.

    Factorability, String Rewriting and Discrete Morse Theory · Examples and counterexamples · 0.81

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