Does the sum of exponents of a symbol in a word of a free group have a specific name in the literature? Let $A$ be a non-empty set and let $F(A)$ be the free group it generates. An element of $F(A)$ is of the form
$$w = a_1^{\varepsilon_1}a_2^{\varepsilon_2}\cdots a_{n-1}^{\varepsilon_{n-1}}a_n^{\varepsilon_n}$$
where $a_i\in A$ and $\varepsilon_i = \pm 1$ for all $i=1,\ldots,n$.
Let us fix a specific element, $b\in A$. We can define the group morphism $w_b:F(A)\to \mathbb{Z}$ where
$$w_b(a_i) = \begin{cases}1 & \text{if }a_i = b\\ 0 & \text{otherwise.} \end{cases}$$
With this definition,
$$w_b(w) = \sum_{\substack{1\leq i\leq n \\ a_i = b}} \varepsilon_i.$$
For example, $w_b(a^2b^{-1}cb^3a^{-1}) = 2$.
It seems like such a morphism would have a name in the literature. If so, what is it?

If $B$ is a subset of $A$, I would call projection from $F(A)$ onto $F(B)$ the morphism $\pi_B: F(A) \to F(B)$ defined, for each $a \in A$, by
$$
\pi_B(a) = 
\begin{cases} 
  a &\text{if $a \in B$}\\
  1 &\text{otherwise}
\end{cases} 
$$
In particular, your morphism would be the projection from $F(A)$ onto $F(b)$.
