Preservation under Reduced Products in Continuous Logic

TL;DR

This paper introduces a fragment of continuous first-order logic preserved under reduced products, extending classical results and characterizing related theories in both classical and metric frameworks.

Contribution

It defines a new fragment of continuous logic analogous to Palyutin formulas and explores its preservation properties under reduced products.

Findings

Fragment preserved under reduced products in both directions

Extension of classical results on complete theories and stability

Characterization of Palyutin sentences and theories in classical and metric logic

Abstract

We introduce a fragment of continuous first-order logic, analogue of Palyutin formulas (or h-formulas) in classical model theory, which is preserved under reduced products in both directions. We use it to extend classical results on complete theories which are preserved under reduced product and their stability. We also characterize the set of Palyutin sentences, Palyutin theories and other related fragments in terms of their preservation properties, both in the classical setting and the metric one.