On the Starheight of Some Rational Subsets Closed under Partial Commutations

Abstract

Let A be a finite alphabet, A * the free monoid it generates, and θ = θ −1 ⊆ A × A a relation of partial commutations. Denote by ≍ the congruence on the free monoid generated by the relators ab ≍ ba for all ( a , b )∈θ. It has been shown that if a word w ∈ A * contains all letters of A then [ w *] = { u ∈ A *| u ≍ w n for some n ≥ 0} is a rational subset if and only if the graph of the complement θ of θ is connected. We prove:(1) if θ is not connected then [ w *] has starheight one (2) if θ and θ are connected then for all integers n there exists a word w∈A* of length O(n) such that [w*] is a rational subset of starheight n.