Let $(G,\cdot)$ be a semigroup, that is, a set $G$ together with a function $\cdot\colon G\times G \to G$ so that for any $g_1,g_2,g_3 \in g$ we have $$\cdot(g_1,\cdot(g_2,g_3)) = \cdot(\cdot(g_1,g_2),g_3)$$ where we denote $\cdot(g_1,g_2)$ by $g_1\cdot g_2$.
I want to see how we can define a function $f\colon G^n\to G$ for each $n \in \mathbb{N}$ so that
$f(g_1,g_2) = g_1\cdot g_2$,
$f(g_1,...,g_n, g_{n+1}) = f(g_1,...,g_n)\cdot g_{n+1}$.
Obviously, a recursion theorem of some form is need to be used here. However, I don't see how we can apply it formally instead of just handwaving away and saying "define by induction".