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It is well-known that some of the most basic properties of words, like the commutativity ($xy=yx$) and the conjugacy ($xz=zy$), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation $x^m{y}^n=z^p$ has only periodic solutions in a free monoid, that is, if $x^m{y}^n=z^p$ holds with integers $m,n,p\ge 2$, then there exists a word $w$ such that $x,y,z$ are powers of $w$. This result, which received a lot of attention, was first proved by Lyndon and...

It is well-known that some of the most basic properties of words, like the commutativity () and the conjugacy (), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation has only periodic solutions in a free monoid, that is, if holds with integers , then there exists a word such that are powers of . This result, which received a lot of attention, was first proved by Lyndon and Schützenberger...