Reducts of Hrushovski's constructions of a higher geometrical arity

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 $\mathbb{M}^{clq}$.