Pseudofinite H-structures and groups definable in supersimple H-structures

TL;DR

This paper investigates properties of H-structures, showing how to construct pseudo-finite H-structures from finite structures and proving the non-existence of new definable groups in supersimple H-structures of SU-rank one.

Contribution

It introduces a construction method for pseudo-finite H-structures based on asymptotic classes and establishes the absence of new definable groups in certain supersimple H-structures.

Findings

H-structures can be constructed as ultraproducts of finite structures.

No new definable groups appear in supersimple H-structures of SU-rank one.

Construction preserves pseudo-finiteness.

Abstract

In this paper we explore some properties of H-structures. We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudo-finiteness. That is, the H-structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU-rank one, there are no new definable groups in H-structures.