Modular functionals and perturbations of Nakano spaces

TL;DR

This paper advances the model theory of Nakano spaces by characterizing isometric embeddings, proving definability of the modular functional, and establishing stability and categoricity under perturbations.

Contribution

It provides new results on the structure, definability, and stability of Nakano spaces, addressing open questions from prior research.

Findings

Isometric embeddings in Nakano spaces are simple and rigid in dimension two and above.

The modular functional is definable in the Banach lattice language.

Atomless Nakano spaces are stable, $eth_0$-categorical, and model complete under small perturbations.

Abstract

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin \cite{Poitevin:PhD}. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and that complete theories of atomless Nakano spaces are model complete. We also show that up to arbitrarily small perturbations of the exponent Nakano spaces are $\aleph_0$-categorical and $\aleph_0$-stable. In particular they are stable.