Idempotent Generated Endomorphisms of an Independence Algebra

TL;DR

This paper provides a direct proof that singular endomorphisms of finite rank independence algebras can be expressed as products of idempotent endomorphisms with the same rank.

Contribution

It offers a straightforward proof of a known result regarding the structure of singular endomorphisms in independence algebras.

Findings

Singular endomorphisms decompose into idempotent factors.

The rank of the endomorphism equals the rank of each idempotent factor.

The proof simplifies understanding of endomorphism structure in independence algebras.

Abstract

The aim of this note is to give a direct proof for the following result proved by Fountain and Lewin: {\em Let $\alg$ be an independence algebra of finite rank and let $a$ be a singular endomorphism of $\alg $. Then $a=e_1... e_n$ where $e^2_i=e_i$ and $rank(a)=rank(e_i)$.}