Say $A$ is an algebra with generating set $\{a_i\}_{i\in I}$, possibly infinite. I want to check that for my specific algebra, that some map $f:A\to A$ is an endomorphism. Can one check this on generators? If so how?
I mean to say, if one can show $f(a_ia_j)=f(a_i)f(a_j)$, for each pair $i,j\in I$, is it true that $f$ is an endomorphism of algebras? I can't see how to move from $f(a_ia_ja_k)=f(a_i)f(a_ja_k)$ for example, which feels like it should be true for some reason.