Deciding Conjugacy of a Rational Relation

Abstract

A rational relation is conjugate if every pair of related words are conjugates. It is shown that checking whether a rational relation is conjugate is decidable. For this, we generalise the Lyndon-Sch\"utzenberger's theorem from word combinatorics. A consequence of the generalisation is that a set of pairs generated by a sumfree rational expression is conjugate if and only if there is a word witnessing the conjugacy of all the pairs