Equations in Words

This chapter discusses equations in words. The notion of an equation in words has been introduced in the framework of group theory. The concept is related to much older ones such as the presentations of groups. Solving an equation in words in the free monoid is a particular case of a unification problem. Indeed, a word on the alphabet A can be viewed as a term in the free algebra on the set A of variables with one function symbol subject to the law of associativity. A solution of an equation ( u, v ) in the free monoid A * is just the same as a unifier of the terms corresponding to u and v . It explains that instead of concentrating on the set of solutions of a given equation, one may consider a fixed morphism s : A* → B* and look for the set of equations ( u, v ) satisfied by s . This is of course the rational binary relation of the form s −1 s .

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