On cancellation properties of languages which are supports of rational power series (Q1066677)

We show two properties of the languages which are supports of noncommutative rational power series: 1. Each such language L possesses a finite test set; that is, there exists a finite language \(K\subset L\) such that whenever two morphisms into a free monoid coincide on K, then they coincide on L. This is a special case of the Ehrenfeucht conjecture, which was given meanwhile a complete positive (and simple) answer: see e.g. \textit{D. Perrin}: ''On the solution of Ehrenfeucht's conjecture'' [Bull. Eur. Assoc. Theor. Comput. Sci. 2]. If two complementary languages are both supports, then they are regular.