A star-height problem in free monoids with partial commutations

Abstract

Let A be a finite alphabet and let ∼ be a congruence on the free monoid A* induced by a set of partial commutations θ=θ^R ⊂ AxA. It has been shown that if w ε A* contains all letters of A then [w*]={u ε A* | u ∼ wⁿ for some n ≥ 0} is a rational set iff the graph of the complement $\overline \theta$ of θ is connected. We prove: 1) if θ is not connected then [w*] has starheight one, 2) if θ and $\overline \theta$ are connected there exist words w ε A* such that [w*] is a rational set of arbitrary starheight.