Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with n≥2), there is a natural construction of a semigroup of transformations ⟨D⟩. For any edge (a, b) of D, let a→b be the idempotent of rank n−1 mapping a to b and fixing all vertices other than a; then, define ⟨D⟩ to be the semigroup generated by a→b for all (a,b)∈E(D). For α∈⟨D⟩, let ℓ(D,α) be the minimal length of a word in E(D) expressing α. It is well known that the semigroup Sing_n of all transformations of rank at most n−1 is generated by its idempotents of rank n−1. When D=K_n is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ℓ(K_n,α), for any α∈⟨K_n⟩=Sing_n; however, no analogous non-trivial results are known when D≠K_n. In this paper, we characterise all simple digraphs D such that either ℓ(D,α) is equal to Howie–Iwahori’s formula for all α∈⟨D⟩, or ℓ(D,α)=n−fix(α) for all α∈⟨D⟩, or ℓ(D,α)=n−rk(α) for all α∈⟨D⟩. We also obtain bounds for ℓ(D,α) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n−1 of Sing_n). We finish the paper with a list of conjectures and open problems