Lengths of words in transformation semigroups generated by digraphs

Abstract

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= Kn is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ℓ(Kn, α) , for any α∈ ⟨ Kn⟩ = Sing n; however, no analogous non-trivial results are known when D≠ Kn. 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.