I've just come across the definition of the n-ary derived operation, namely that starting with an operational type $(\Omega, \alpha)$, set $ X_n = (x_1, ... , x_n) $ and $ \Omega$-structure $A$, we can define a function $ t_A : A^n \to A $ for each $t \in FX_n $ inductively as follows:
if $ t = x_i \ (1 \leq i \leq n) $, then $t_A$ is the projection map onto the ith factor.
if $ t = \omega t_1 ... t_m$ where $ \alpha(\omega) = m $, then $t_A$ is the composite map $ \omega_A \circ ((t_1)_A, ... , (t_m)_A) : A^n \to A^m \to A $.
In particular, if $t$ is the term $\omega x_1 ... x_n $ where $ n = \alpha(\omega) $, then $ t_A = \omega_A$.
Now I'm trying to make this definition more concrete, by interpreting it in a group theoretical context. Take $ \Omega = (m, i, e) $, where $m, i, e$ are the binary, unary and nullary group operations respectively. Now, I'm getting confused about the roles that $ X_n$ and $A$ play here. Which should I take to be the set of elements of the group? What should the other be interpreted as (if applicable)?