Subwords and Power-Free Words are not Expressible by Word Equations

We consider several open problems of Karhumäki, Mignosi, and Plandowski, cf. [KMP], concerning the expressibility of languages and relations as solutions of word equations. We show first that the (scattered) subword relation is not expressible. Then, we consider the set of k-power-free finite words and solve it negativelly for all nontrivial integer values of k. Finally, we consider the Fibonacci finite words. We do not solve the problem of the expressibility of the set of these words but prove that a negative answer (as believed) cannot be given using the tools in [KMP].