To simplify matters, assume we have a commutative group $(X,\cdot,1)$ with uncountable $X$. For commutative groups, applications of elements $x_i\in X$ don’t care about order, and we can simply count the number of times each element of $X$ is applied. This means that if we don’t apply any element in $X$ twice, (e.g. $a\cdot b \cdot c$ but not $a\cdot a\cdot c$), then we can simply associate with each application of elements, a set $Y\subseteq X$ (This is just to simplify the question but not essential)
This means we can represent the $\cdot:X\times X\to X$ operator instead as a function $\square : \mathcal P (X) \to X$.
Normally, this set inputted in $\square$ will be finite or at least countable, because when we do applications of elements of $X$ we list this as a sequence of applications.
But the type signature of $\square$ suggests that we can also input uncountable subsets of $X$. Is there a theory about this? This would be an uncountable application of an operator.