On Weak Elimination of Hyperimaginaries and its Consequences

TL;DR

This paper investigates weak elimination of hyperimaginaries in simple theories, establishing a characterization via dependence-witnessed-by-imaginaries, and explores its implications for forking, thorn-forking, and stability in finite SU-rank theories.

Contribution

It introduces a new characterization of WEHI through DWIP and links WEHI to the equivalence of forking and thorn-forking, extending previous results.

Findings

In simple theories with WEHI, forking and thorn-forking coincide.

The equivalence of independence and thorn-independence implies WEHI.

Stable forking is demonstrated in 1-based theories of finite SU-rank with WEHI.

Abstract

We analyze the notion of weak elimination of hyperimaginaries (WEHI) in simple theories. A key observation in the analysis is a characterization of WEHI in terms of forking dependence -- a condition we dub dependence-witnessed-by-imaginaries (DWIP). Generalizing results of [1] and [3], we show that in a simple theory with WEHI, forking and thorn-forking coincide. We also show that, conversely, the equivalence of independence and thorn-independence is (almost) sufficient for WEHI. Thus, the WEHI and the statement independence = thorn-independence are morally equivalent. As a further application of our technology, we demonstrate stable forking for 1-based theories of finite SU-rank that have WEHI.