Notation with ultrafilter?

Question

I am reading about ultrafilter in logic book and I see a notation where we have a structure $S$ in a language $L$, an ultrafilter $U$, and the notation is $S^{U}$ , but I don't understand what this mean. I try to search elsewhere but have no luck. Please explain this notation?

Answer 1

This is the ultrapower of $S$. Equivalent notation includes $$\prod_{i\in I} S \, / \, U, \text{or } S^I/\,U$$ (If $U$ is an ultrafilter on $I$).