Let
\begin{align} S(f)(x) &= f(x + 1) \\ R_a(f)(x) &= \begin{cases} a & x = 0 \\ f(x) & \text{otherwise} \end{cases} \end{align}
What is the presentation of the monoid generated by $\{S, S^{-1}\} \cup \{R_a \mid a \in A\}$ under function composition? Is it the following?
\begin{align} S S^{-1} &= 1 \\ S^{-1} S &= 1 \\ R_a R_b &= R_a \\ (S^n \cdot R_a) R_b &= R_b (S^n \cdot R_a) & n \neq 0 \end{align}
where $a \cdot b = a b a^{-1}$ denotes conjugation. The last relation entails
\begin{align} (S^m \cdot R_a) (S^n \cdot R_b) &= (S^n \cdot R_b) (S^m \cdot R_a) & m \neq n \end{align}
Are there other relations? Can it be made finitely related?
Edit: I think the following is a normal form for this monoid:
\begin{align} &S^m \prod_i (S^{n_i} \cdot R_{a_i}) & i < j \Rightarrow n_i < n_j \end{align}