The isomorphism theorem for linear fragments of continuous logic

Abstract

The ultraproduct construction is generalized to $p$-ultramean constructions ($1⩽p<\infty$) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments ${\mathcal{L}}^p$ of continuous logic. A powermean variant of Keisler-Shelah isomorphism theorem is proved for ${\mathcal{L}}^p$. It is then proved that ${\mathcal{L}}^p$-sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.