New decidable upper bound of the second level in the Straubing-Thérien concatenation hierarchy of star-free languages

In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Therien concatenation hierarchy of star-free languages. In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new upper bound. We also prove that this new upper bound is incomparable with the previous upper bound.

[1]  Pascal Weil,et al.  A conjecture on the concatenation product , 2001, RAIRO Theor. Informatics Appl..

[2]  Denis Thérien,et al.  Classification of Finite Monoids: The Language Approach , 1981, Theor. Comput. Sci..

[3]  Jorge Almeida,et al.  Finite Semigroups and Universal Algebra , 1995 .

[4]  Mustapha Arfi.  Polynomial operations and hierarchies of concatenation (in French) , 1991 .

[5]  Benjamin Steinberg,et al.  The q-theory of Finite Semigroups , 2008 .

[6]  Jean-Eric Pin,et al.  A Reiterman theorem for pseudovarieties of finite first-order structures , 1996 .

[7]  Mustapha Arfi.  Opérations polynomiales et hiérarchies de concaténation , 1991, Theor. Comput. Sci..

[8]  Howard Straubing.  Semigroups and Languages of Dot-Depth 2 , 1986, ICALP.

[9]  V. Molchanov.  Nonstandard characterization of pseudovarieties , 1995 .

[10]  Jorge Almeida,et al.  A counterexample to a conjecture concerning concatenation hierarchies , 2009, Inf. Process. Lett..

[11]  David F. Cowan,et al.  Inverse Monoids of dot-Depth Two , 1993, Int. J. Algebra Comput..

[12]  Manfred Kufleitner.  The Height of Factorization Forests , 2008, MFCS.

[13]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[14]  Jean-Éric Pin,et al.  Algebraic tools for the concatenation product , 2003, Theor. Comput. Sci..

[15]  Jean-Eric Pin,et al.  Profinite Semigroups, Mal'cev Products, and Identities☆ , 1996 .

[16]  Howard Straubing,et al.  On a Conjecture Concerning Dot-Depth Two Languages , 1992, Theor. Comput. Sci..

[17]  Jan Reiterman,et al.  The Birkhoff theorem for finite algebras , 1982 .

[18]  Thomas Sudkamp,et al.  Languages and Machines , 1988 .

[19]  Marcel Paul Schützenberger,et al.  On Finite Monoids Having Only Trivial Subgroups , 1965, Inf. Control..

[20]  Pascal Weil,et al.  Polynomial closure and unambiguous product , 1995, Theory of Computing Systems.

[21]  Ondrej Klíma,et al.  Polynomial Operators on Classes of Regular Languages , 2009, CAI.

[22]  V. Molchanov.  On nonstandard axiomatization of elementarily nonaxiomatizable classes of discrete algebraic systems , 1999 .

[23]  Howard Straubing,et al.  FINITE SEMIGROUP VARIETIES OF THE FORM V,D , 1985 .