# A conjecture on the concatenation product

@article{Pin2001ACO, title={A conjecture on the concatenation product}, author={Jean-{\'E}ric Pin and Pascal Weil}, journal={RAIRO Theor. Informatics Appl.}, year={2001}, volume={35}, pages={597-618} }

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure --- this operation corresponds to passing to the upper level in any concatenation hierarchy ---. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case…

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## References

SHOWING 1-10 OF 77 REFERENCES

### Bridges for Concatenation Hierarchies

- MathematicsICALP
- 1998

This article solves positively a question raised in 1985 about concatenation hierarchies of rational languages, which are constructed by alternating boolean operations and concate-nation products, and establishes a simple algebraic connection between the Straubing-Therien hierarchy and the group hierarchy.

### Products of group languages

- Computer ScienceFCT
- 1985

The aim of this paper is to study the hierarchy whose level 0 consists of all group languages, and the union U of all levels of the hierarchy is the closure of group languages under product and boolean operations.

### THE WREATH PRODUCT PRINCIPLE FOR ORDERED SEMIGROUPS

- Mathematics
- 2002

ABSTRACT Straubing's wreath product principle provides a description of the languages recognized by the wreath product of two monoids. A similar principle for ordered semigroups is given in this…

### Product of Group Languages

- Mathematics
- 1985

The aim of this paper is to study the concatenation hierarchy whose level 0 consists of all group languages. The union of all the levels of this hierarchy is the closure of group languages under…

### Polynomial Closure and Topology

- MathematicsInt. J. Algebra Comput.
- 2000

This paper studies the pseudovariety of monoids corresponding to the variety of languages generated by the polynomial closure of the varieties of H-languages and the Pseudo-Pseudo-Monoid equivalent corresponding to ordered monoids, and obtains a basis of ordered pseudoidentities for such positive varieties.

### Hiérarchies de Concaténation

- MathematicsRAIRO Theor. Informatics Appl.
- 1984

New hierarchies of varieties of languages (based on the concatenation product) are defined and an algebraic description of the corresponding hierarchiesof varieties of semigroups is given.

### Ash's Type II Theorem, Profinite Topology and Malcev Products: Part I

- MathematicsInt. J. Algebra Comput.
- 1991

It is shown that the largest local complexity function in the sense of Rhodes and Tilson is computable and connections with the complexity theory of finite semigroups are given.

### Free Profinite ℛ-Trivial Monoids

- MathematicsInt. J. Algebra Comput.
- 1997

The structure of semigroups of implicit operations on R, the pseudovariety of all ℛ-trivial semig groups, is described by means of labeled ordinals and of labeled infinite trees of finite depth.