Compositions of $n$-homomorphisms

TL;DR

This paper explores the properties and compositions of n-homomorphisms between rings, revealing algebraic structures and combinatorial proofs for their additive and multiplicative behaviors.

Contribution

It generalizes the concept of n-homomorphisms to arbitrary rings and establishes their compositional properties with combinatorial proofs.

Findings

Sum of an n-homomorphism and an m-homomorphism is an (n+m)-homomorphism.

Composition of an n-homomorphism and an m-homomorphism is an nm-homomorphism.

Proofs are entirely combinatorial.

Abstract

We study $n$-homomorphisms in the sense of Khudaverdian--Voronov, but generalized to maps from arbitrary rings to arbitrary commutative rings. We show that the sum of an $n$-homomorphism and an $m$-homomorphism is an $\left( n+m\right) $-homomorphism, and that the composition of an $n$-homomorphism and an $m$-homomorphism is an $nm$-homomorphism. The proofs are entirely combinatorial.