## Solutions principales et rang d’un système d’équations avec constantes dans le monoide libre.(French)Zbl 0545.20046

In order to give a far-reaching common generalization of Lentin’s and Makanin’s results concerning equations without and with constants, the author defines a system of equations over a finite alphabet $$E\cup C (E\cap C=\emptyset)$$ to be a set of quadruples $$(e_ i,e'\!_ i,E,C)$$, $$e_ i,e'\!_ i\in(E\cup C)^*$$, and a solution of this system to be a morphism $$\alpha:(E\cup C)^*\to(A\cup C)^*$$ where A is arbitrary finite such that $$A\cap C=\emptyset$$, $$\alpha$$ acts identically on C, and $$\alpha e_ i=\alpha e'\!_ i$$. All essential results carry over to this case. In particular, every solution can be derived from a unique principal solution, and the calculation of the latter ones is equivalent (at least in the case of a finite system) to finding the principal solutions of some single equation of the same rank. Thus, the rank of a system can be determined, too.
Reviewer: G.Pollák

### MSC:

 20M05 Free semigroups, generators and relations, word problems 20M35 Semigroups in automata theory, linguistics, etc.

### Keywords:

system of equations; principal solutions; rank
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### References:

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