Submission date: 17 June 2021

Abstract:

In this paper we study the Lascar group over a hyperimaginary e. We verify that various results about the group over a real set still hold when the set is replaced by e. First of all, there is no written proof in the available literature that the group over e is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book, Simplicity Theory [6, 5.1.14-15] and produce a counterexample. On the other, we extend Newelski's Theorem in The diameter of a Lascar strong type [12] that `a G-compact theory over a set has a uniform bound for the Lascar distances' to the hyperimaginary context. Lastly, we supply a partial positive answer to a question raised in The relativized Lascar groups, type-amalgamations, and algebraicity [4, 2.11], which is even a new result in the real context.

Mathematics Subject Classification: 03C60 (Primary) 54H11 (Secondary)

Keywords and phrases: Lascar group, hyperimaginary, strong types, G-compactness

Full text arXiv 2106.09200: pdf, ps.