Maximal stable quotients of invariant types in NIP theories

TL;DR

This paper establishes the existence of a maximal stable quotient of an invariant type in NIP theories, extending the concept of stable quotients from definable groups to types using automorphism groups.

Contribution

It introduces a new result showing the existence of a finest relatively type-definable stable quotient for invariant types in NIP theories, generalizing previous work on definable groups.

Findings

Existence of a finest relatively type-definable stable quotient for invariant types.

Extension of stable quotient concepts from groups to types in NIP theories.

Method adapts automorphism group techniques from prior research.

Abstract

For a NIP theory $T$, a sufficiently saturated model $\mathfrak{C}$ of $T$, and an invariant (over some small subset of $\mathfrak{C}$) global type $p$, we prove that there exists a finest relatively type-definable over a small set of parameters from $\mathfrak{C}$ equivalence relation on the set of realizations of $p$ which has stable quotient. This is a counterpart for equivalence relations of the main result of the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay which shows the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in the paper "On first order amenability" by E. Hrushovski, K. Krupinski, and A. Pillay.