# The Complexity of Homomorphism Factorization

## TL;DR

This paper studies the computational complexity of algebra homomorphism factorization, proving NP-completeness in general but identifying polynomial-time solvable cases for specific algebra classes.

## Contribution

It establishes NP-completeness of the homomorphism factorization problem and identifies classes where the problem is solvable in polynomial time.

## Key findings

- Homomorphism factorization problems are NP-complete.
- Polynomial-time solutions exist for Boolean algebras, vector spaces, G-sets, and abelian groups.
- Developed a technique for compatible restrictions on homomorphisms.

## Abstract

We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, L, and take as input an algebra homomorphism f between two finite L-algebras X and Z, along with an intermediate finite L-algebra Y. The decision problem asks whether there are homomorphisms g from X to Y and h from Y to Z such that f = hg. We show that these Homomorphism Factorization Problems are NP-complete. We also develop a technique for producing compatible restrictions on homomorphisms, and show that Homomorphism Factorization Problems have polynomial time instances for finite Boolean algebras, finite vector spaces, finite G-sets, and finite abelian groups.