n-Homomorphisms

Let $\mathcal{A}$ and $\mathcal{B}$ be two (complex) algebras. A linear map $\varphi :\mathcal{A}\to \mathcal{B}$ is called $n$-homomorphism if $\phi(a_{1}… a_{n})=\phi(a_{1})…\phi(a_{n})$foreach$a_{1},…,a_{n}\in{\mathcal A}.$ In this paper, we investigate $n$-homomorphisms and their relation to homomorphisms. We characterize $n$-homomorphisms in terms of homomorphisms under certain conditions. Some results related to continuity and commutativity are given as well.