Compositions of $n$-homomorphisms

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.

28 pages, partly expository. A crucial lemma was found and proved by GPT-5.4, but all of the writing is my own