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 $PG({\mathbb{M}}_3)\ncong PG({\mathbb{M}}_4)$. 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 Fraïssé-Hrushovski limit incorporating into the construction non-eliminable imaginary sorts in ${\mathbb{M}}^{clq}$.