A conjecture on the concatenation product
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 6, pp. 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. 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.

DOI : https://doi.org/10.1051/ita:2001134
Classification : 20M07,  68Q45,  20M35
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Pin, Jean-Eric; Weil, Pascal. A conjecture on the concatenation product. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 6, pp. 597-618. doi : 10.1051/ita:2001134. http://www.numdam.org/articles/10.1051/ita:2001134/

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