A conjecture on the concatenation product
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, p. 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
@article{ITA_2001__35_6_597_0,
     author = {Pin, Jean-Eric and Weil, Pascal},
     title = {A conjecture on the concatenation product},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {6},
     year = {2001},
     pages = {597-618},
     doi = {10.1051/ita:2001134},
     zbl = {1011.20054},
     mrnumber = {1922298},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2001__35_6_597_0}
}
Pin, Jean-Eric; Weil, Pascal. A conjecture on the concatenation product. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, pp. 597-618. doi : 10.1051/ita:2001134. http://www.numdam.org/item/ITA_2001__35_6_597_0/

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