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Some results on the dot-depth hierarchy

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Résumé

Dans ce travail, nous poursuivons l'étude de la décidabilité de la hiérarchie de concaténation dite “dot-depth”. Nous donnons une borne inférieure effective de la dot-depth d'un monoïde apériodique. Notre outil principal pour ce résultat est l'étude d'une certaine opération sur les variétés de monoïdes finis en termes de produit de Mal'cev. Nous prouvons aussi l'égalité de deux variétés décidables, dont on sait qu'elles continennent tous les monoïdes de dot-depth deux. Enfin, nous restreitgnant aux monoïdes inversifs, nous, montrons que la classe des monoïdes inversifs de dot-depth deux est localement finie.

Abstract

In this paper we pursue the study of the decidability of the dot-depth hierarchy. We give an effective lower bound for the dotdepth of an aperiodic monoid. The main tool for this is the study of a certain operation on varieties of finite monoids in terms of Mal'cev product. We also prove the equality of two decidable varieties which were known to contain all dot-depth two monoids. Finally, we restrict our attention to inverse monoids, and we prove that the class of inverse dot-depth two monoids is locally finite.

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Authors and Affiliations

  1. LITP-Institut Blaise Pascal, 4 place Jussieu, 75252, Paris Cedex 05, France

    Pascal Weil

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  1. Pascal Weil
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Additional information

Communicated by Jean-Eric Pin

Partial support form the following is gratefully acknowledged: PRC “Mathématiques et Informatique”, ESPRIT-BRA project 3166 “ASMICS”, NSF grant DMS-8702019.

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Weil, P. Some results on the dot-depth hierarchy. Semigroup Forum 46, 352–370 (1993). https://doi.org/10.1007/BF02573578

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  • DOI: https://doi.org/10.1007/BF02573578

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