An ab initio construction of a geometry

We show that the geometry of Hrushovski's ab initio construction for a single $n$-ary relation not-permitting dependent sets of size less than $n$, when restricted to $n$-tuples, can be itself constructed as a Hrushovski construction.Comment: 8 pages. Theorem 4.12 was removed due to a gap in the proo

Reduction relations between non-collapsed ab initio Hrushovski constructions of varying degrees of symmetry

Denote Hrushovski's non-collapsed ab initio construction for an $n$-ary relation by $\mathbb{M}$ and the analogous construction for a symmetric $n$-ary relation by $\mathbb{M}^{\sim}$. We show that $\mathbb{M}$ is isomorphic to a proper reduct of $\mathbb{M}^{\sim}$ and vice versa, and that the combinatorial geometries associated with both structures are isomorphic.Comment: 33 page

Reducts of Hrushovski's constructions of a higher geometrical arity

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 $\mathbb{M}^{clq}$

The generic flat pregeometry

We examine the first order structure of pregeometries of structures built via Hrushovski constructions. In particular, we show that the class of flat pregeometries is an amalgamation class such that the pregeometry of the unbounded arity Hrushovski construction is precisely its generic. We show that the generic is saturated, provide an axiomatization for its theory, show that the theory is $\omega$-stable, and has quantifier-elimination down to boolean combinations of $\exists\forall$-formulas. We show that the pregeometries of the bounded-arity Hrushovski constructions satisfy the same theory, and that they in fact form an elementary chain