A Tame Generic Structure with Non-Algebraic Geometric Closure

TL;DR

This paper introduces paracollapsed structures, a new class of infinite structures with unique geometric closure properties, expanding the understanding of model-theoretic structures while maintaining decidability.

Contribution

It develops a procedure to construct paracollapsed structures using Hrushovski's amalgamation, demonstrating their properties and significance in model theory.

Findings

Paracollapsed structures have the strict order property.

They exhibit TP2.

The resulting theories are decidable.

Abstract

By providing a procedure to apply Hrushovski's amalgamation method to the setting of classes of infinite structures, we introduce the notion of \textit{paracollapsed} structures. We show that this approach provides existentially closed generic structures in which the geometric closure is not included in the algebraic closure while the resulting theory is decidable. We show that paracollapsed structures have the strict order property and $\text{TP}_2$.