On post correspondence problem for letter monotonic languages

We prove that for given morphisms g,h:{a1,a2,…,an}→B∗, it is decidable whether or not there exists a word w in the regular language such that g(w)=h(w). In other words, we prove that the Post Correspondence Problem is decidable if the solutions are restricted to be from this special language. This yields a nice example of an undecidable problem in integral matrices which cannot be directly proved undecidable using the traditional reduction from the Post Correspondence Problem.