Lengths of words in transformation semigroups generated by digraphs

TL;DR

This paper characterizes when the length of words in semigroups generated by digraphs matches known formulas, provides bounds for acyclic digraphs, and explores special cases like strong tournaments.

Contribution

It offers a complete characterization of digraphs where word lengths follow specific formulas and analyzes length bounds for acyclic digraphs and strong tournaments.

Findings

Characterization of digraphs with length formulas

Tight upper bounds for acyclic digraphs

Analysis of strong tournaments and conjectures

Abstract

Given a simple digraph $D$ on $n$ vertices (with $n\ge2$), there is a natural construction of a semigroup $\langle D\rangle$ associated with $D$. For any edge $(a,b)$ of $D$, let $a\to b$ be the idempotent of defect $1$ mapping $a$ to $b$ and fixing all vertices other than $a$; then define $\langle D\rangle$ to be the semigroup $\langle a\to b:(a,b)\in E(D)\rangle$. For $\alpha \in \langle D \rangle$, let $\ell(D,\alpha)$ be the minimal length of a word in $E(D)$ expressing $\alpha$. When $D=K_n$ is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate $\ell(K_n,\alpha)$, for any $\alpha \in \langle K_n \rangle = \text{Sing}_n$; however, no analogous nontrivial results are known when $D \neq K_n$. In this paper, we characterise all simple digraphs $D$ such that either $\ell(D,\alpha)$ is equal to Howie-Iwahori's formula for all $\alpha \in \langle D \rangle$, or $\ell(D,\alpha) = n - \text{fix}(\alpha)$ for all $\alpha \in \langle D \rangle$, or $\ell(D,\alpha) = n - \text{rk}(\alpha)$ for all $\alpha \in \langle D \rangle$. When $D$ is an acyclic digraph and $\alpha \in \langle D \rangle$, we find a tight upper bound for $\ell(D,\alpha)$. Finally, we study the case when $D$ is a strong tournament (which corresponds to a smallest generating set of idempotents of defect $1$ of $\text{Sing}_n$), and we propose some conjectures.