Classification over a predicate -- the general case. Part I -- structure theory

TL;DR

This paper develops a comprehensive structure theory for first-order theories stable over a monadic predicate, introducing key concepts like definability, independence, and stability properties.

Contribution

It provides the first systematic development of structure theory for such theories, including stability implications, independence notions, and amalgamation results.

Findings

Types over stable sets are quantifier free definable.

An independence notion with specific properties is introduced.

Types orthogonal to the predicate are generically stable.

Abstract

We begin a systematic development of structure theory for a first order theory, which is stable over a monadic predicate. We show that stability over a predicate implies quantifier free definability of types over stable sets, introduce an independence notion and explore its properties, prove stable amalgamation results, and show that every type over a model, orthogonal to the predicate, is generically stable.