The isomorphism theorem for linear fragments of continuous logic

TL;DR

This paper extends ultraproducts to p-ultramean constructions in continuous logic, proving a Keisler-Shelah isomorphism theorem for linear fragments and characterizing sentences preserved by these constructions.

Contribution

It introduces p-ultramean constructions for continuous logic and establishes a corresponding isomorphism theorem, expanding the understanding of linear fragments in continuous logic.

Findings

Generalized ultraproducts to p-ultramean constructions.

Proved a powermean version of Keisler-Shelah isomorphism theorem.

Characterized sentences preserved by p-ultramean constructions.

Abstract

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