A Tame Generic Structure with Non-Algebraic Geometric Closure
Somaye Jalili, Massoud Pourmahdian, Ali N. Valizadeh
TL;DR
The paper develops the paracollapsed Hrushovski framework to obtain existentially closed generics whose geometric closure is not contained in the algebraic closure, yet whose theory remains decidable. It introduces a μ-parameterized paracollapsed class and a corresponding model-complete theory $\mathbb{T}_{\mu}$, built as a model companion to an inductive base theory via intrinsic semigenericity axioms. The authors prove consistency, completeness, and quantifier elimination for $\mathbb{T}_{\mu}$, while showing that the theory exhibits $SOP$ and $TP_2$ through explicit constructions and formulas, including $SOP_n$ for all $n\ge3$. These results illustrate a tunable balance between tameness and wildness in Hrushovski-type generics and pave the way for further refinements that might yield non-simple theories without certain strong order properties. Overall, the work contributes a decidable, yet nontrivially ordered, generic structure in the landscape between stability and instability.
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 $\text{TP}_2$.