Finitely Generated Free Group to Finitely Generated Free Monoid

Question

Let $F_n$ be the free group on $n$ generators $u_1,...,u_n$ and $M_n$ the free monoid on $n$ generators $v_1,...,v_n$. Would $u_i \to v_i$ and $u_i^{-1} \mapsto v_i$ extend to a well-defined map that is something like a homomorphism?

Answer 1

This doesn't work, and more generally if $G$ is any group and $M$ is a free monoid, the only (monoid) homomorphism from $G$ to $M$ is the map that sends every element of $G$ to $1$. Indeed, suppose $f:G\to M$ is a homomorphism and let $g\in G$. Then $f(g)f(g^{-1})=f(1)=1$. But the only way two elements of $M$ can have product $1$ is if both elements are $1$ (for instance, because every element of $M$ can uniquely be written as a word in the free generators and the only way to concatenate two words to get an empty word is if the two words are also empty). Thus $f(g)=1$.