A Proof of the Factorization Forest Theorem

TL;DR

This paper proves that for any homomorphism from a free monoid to a finite semigroup, a factorization forest of bounded height exists, with the proof utilizing Green's relations.

Contribution

It introduces a new proof of the Factorization Forest Theorem using Green's relations, establishing a bound on the height of the forest.

Findings

Existence of factorization forests with height ≤ 3|S| for any homomorphism to a finite semigroup

The proof employs Green's relations to establish the bound

Provides a constructive approach to factorization forests in algebraic structures

Abstract

We show that for every homomorphism $\Gamma^+ \to S$ where $S$ is a finite semigroup there exists a factorization forest of height $\leq 3 \abs{S}$. The proof is based on Green's relations.