Affine logic with the integration operator

Abstract

Affine continuous logic is extended to affine integration logic. Affine compactness theorem is proved by both the ultramean construction and Henkin's method. Also, a proof system and a completeness theorem are given. An appropriate variant of the Keisler-Shelah isomorphism theorem holds in this setting. This helps us to characterize non-forking extensions in affine stable theories by means of the notion of elementary embedding in the expanded logic.