Indifference to symmetry in Hrushovski's ab initio construction

Abstract

Denote Hrushovski's non-collapsed ab initio construction for an $n$-ary relation by $\mathcal{M}_{\not\sim}$ and the analogous construction for a symmetric $n$-ary relation by $\mathcal{M}_{\sim}$. We show that $\mathcal{M}_{\not\sim}$ is isomorphic to a proper reduct of $\mathcal{M}_{\sim}$ and vice versa, and that the combinatorial pregeometries associated with both structures are isomorphic.