Ultrafilter extensions do not preserve elementary equivalence

Abstract

We show that there exist models ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ such that ${\mathcal{M}}_1$ elementarily embeds into ${\mathcal{M}}_2$ but their ultrafilter extensions $\beta ({\mathcal{M}}_1)$ and $\beta ({\mathcal{M}}_2)$ are not elementarily equivalent.