Reducts of Hrushovski's constructions of a higher geometrical arity
Volume 247 / 2019
Fundamenta Mathematicae 247 (2019), 151-164
MSC: Primary 03C30, 03C45; Secondary 03C13.
DOI: 10.4064/fm645-10-2018
Published online: 31 May 2019
Abstract
Let $\mathbb {M}_n$ denote the structure obtained from Hrushovski’s (non-collapsed) construction with an $n$-ary relation and $\operatorname{PG} (\mathbb {M}_n)$ its associated pregeometry. It was shown by Evans and Ferreira (2011) that $\operatorname{PG} (\mathbb {M}_3)\not \cong \operatorname{PG} (\mathbb {M}_4)$. We show that $\mathbb {M}_3$ has a reduct $\mathbb {M}^{\operatorname{clq} }$ such that $\operatorname{PG} (\mathbb {M}_4)\cong \operatorname{PG} (\mathbb {M}^{\operatorname{clq} })$. To achieve this we show that $\mathbb {M}^{\operatorname{clq} }$ is a slightly generalised Fraïssé–Hrushovski limit incorporating non-eliminable imaginary sorts in $\mathbb {M}^{\operatorname{clq} }$.
