Sur Certains Systèmes d'Équations avec Contraintes dans Un Groupe Libre

Jorge Almeida and Manuel Delgado

Abstract: The main theorem found by Ash in his proof of the type II conjecture is reformulated in terms of the solvability in the free group $F$ of systems of equations of the form $x\,y=z$ (where $y$ appears only once) with constraints given by subsets of the form $g\,H_1\cdots H_n$ where $g\in F$ and the $H_i$ are subgroups of $F$. More precisely, if such a system has no solution for finitely generated subgroups $H_i$, then one may replace each $H_i$ by a subgroup of finite index containing it so that the system remains without solution.

Keywords: Free group; equation; profinite topology; relational morphism; labeled graph; free monoid; finite monoid; pseudovariety.

Classification (MSC2000): 20M07, 20E18, 20E10.

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