# A conjecture on the concatenation product

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 35, Issue: 6, page 597-618
- ISSN: 0988-3754

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topPin, Jean-Eric, and Weil, Pascal. "A conjecture on the concatenation product." RAIRO - Theoretical Informatics and Applications 35.6 (2010): 597-618. <http://eudml.org/doc/92687>.

@article{Pin2010,

abstract = {
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. Another
counterexample, of a different nature, was independently given
recently by Steinberg. Taking these two counterexamples into
account, we propose a modified version of our conjecture and some
supporting evidence for that new formulation. We show in
particular that a solution to our new conjecture would give a
solution of the decidability of the levels 2 of the
Straubing–Thérien hierarchy and of the dot-depth hierarchy.
Consequences for the other levels are also discussed.
},

author = {Pin, Jean-Eric, Weil, Pascal},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {rational languages; star-free languages; concatenation products; concatenation hierarchies; dot-depth hierarchies; pseudovarieties of semigroups; Mal'cev products; varieties of languages; semidirect products},

language = {eng},

month = {3},

number = {6},

pages = {597-618},

publisher = {EDP Sciences},

title = {A conjecture on the concatenation product},

url = {http://eudml.org/doc/92687},

volume = {35},

year = {2010},

}

TY - JOUR

AU - Pin, Jean-Eric

AU - Weil, Pascal

TI - A conjecture on the concatenation product

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 35

IS - 6

SP - 597

EP - 618

AB -
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. Another
counterexample, of a different nature, was independently given
recently by Steinberg. Taking these two counterexamples into
account, we propose a modified version of our conjecture and some
supporting evidence for that new formulation. We show in
particular that a solution to our new conjecture would give a
solution of the decidability of the levels 2 of the
Straubing–Thérien hierarchy and of the dot-depth hierarchy.
Consequences for the other levels are also discussed.

LA - eng

KW - rational languages; star-free languages; concatenation products; concatenation hierarchies; dot-depth hierarchies; pseudovarieties of semigroups; Mal'cev products; varieties of languages; semidirect products

UR - http://eudml.org/doc/92687

ER -

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