Stability in affine logic
Itaï Ben Yaacov, Tomás Ibarlucía
TL;DR
The paper develops a comprehensive stability theory for affine logic, establishing affine analogues of core stability notions such as definability of types, non-forking extensions, and stationarity. It proves that affine stability is preserved under direct integrals of measurable fields, and that stability in extremal models suffices to transfer stability to the whole theory, connecting with stable continuous logic via the affine part and convex realisation completions. A robust independence calculus is built around stable formulas, with a unique non-forking extension and a well-behaved notion of independence, and Lascar-type results are shown for affine theories. The work unifies stability phenomena across affine and continuous logics, and provides tools for analyzing randomisations, extremal models, and disintegration of affine types.
Abstract
We develop foundational aspects of stability theory in affine logic. On the one hand, we prove appropriate affine versions of many classical results, including definability of types, existence of non-forking extensions, and other fundamental properties of forking calculus. Most notably, stationarity holds over arbitrary sets (in fact, every type is Lascar strong). On the other hand, we prove that stability is preserved under direct integrals of measurable fields of structures. We deduce that stability in the extremal models of an affine theory implies stability of the theory. We also deduce that the affine part of a stable continuous logic theory is affinely stable, generalising the result of preservation of stability under randomisations.