Reducts of Hrushovski's constructions of a higher geometrical arity

TL;DR

This paper demonstrates that a reduct of a Hrushovski construction with a ternary relation can produce a pre-geometry isomorphic to that of a construction with a quaternary relation, revealing new structural connections.

Contribution

It introduces a reduct of Hrushovski's construction that aligns the pre-geometry of different arities, using a generalized Fra"issé-Hrushovski limit with non-eliminable imaginaries.

Findings

$PG( ext{M}_4)$ is isomorphic to $PG( ext{M}^{clq})$

$ ext{M}^{clq}$ is a generalized Fra"issé-Hrushovski limit

Reducts can unify geometries of different arities

Abstract

Let $\mathbb{M}_n$ denote the structure obtained from Hrushovski's (non collapsed) construction with an n-ary relation and $PG(\mathbb{M}_n)$ its associated pre-geometry. It was shown by Evans and Ferreira that $PG(\mathbb{M}_3)\not\cong PG(\mathbb{M}_4)$. We show that $\mathbb{M}_3$ has a reduct, $\mathbb{M}^{clq}$ such that $PG(\mathbb{M}_4)\cong PG(\mathbb{M}^{clq})$. To achieve this we show that $\mathbb{M}^{clq}$ is a slightly generalised Fra\"iss\'e-Hrushovski limit incorporating into the construction non-eliminable imaginary sorts in $\mathbb{M}^{clq}$.