Powerset extension of a binary operation of a semigroup

Question

Given a commutative semigroup $(S,\cdot)$, it is possible to define an operator $\underline{\cdot}: \wp(S)\setminus\{\emptyset\} \rightarrow S$ such that $\underline{\cdot}(X) = s_1 \cdot s_2 \cdot \dots s_{\lvert X \rvert}$?

1. If $\lvert X \rvert < \aleph_0$ (finite) then yes, we can apply $\cdot$ recursively on couples: $\underline{\cdot}(X) = s_1 \cdot (s_2 \cdot \dots (s_{\lvert X \rvert -1} \cdot s_{\lvert X \rvert})\dots))$.
2. If $\lvert X \rvert = \aleph_0$ (countably infinite) ?
3. If $\lvert X \rvert > \aleph_0$ (uncountably infinite) ?

Can we apply the method of point 1. also for points 2. and 3.?