A note on Łoś theorem without the Axiom of Choice

We study some topics around Łoś’s theorem without assuming the Axiom of Choice. We prove that Łoś’s fundamental theorem on ultraproducts is equivalent to a weak form that every ultrapower is elementarily equivalent to its source structure. On the other hand, it is consistent that there is a structure $M$ and an ultrafilter $U$ such that the ultrapower of $M$ by $U$ is elementarily equivalent to $M$, but the fundamental theorem for the ultrapower of $M$ by $U$ fails. We also show that weak fragments of the Axiom of Choice, such as the Countable Choice, do not follow from Łoś’s theorem, even assuming the existence of non-principal ultrafilters.