Compositions of $n$-homomorphisms

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 $(n+m)$-homomorphism, and that the composition of an $n$-homomorphism and an $m$-homomorphism is an $nm$-homomorphism. The proofs are entirely combinatorial.