Indifference to symmetry in Hrushovski's ab initio construction

Abstract

Denote Hrushovski's non-collapsed ab initio construction for an n-ary relation by M≁ and the analogous construction for a symmetric n-ary relation by M∼. We show that M≁ is isomorphic to a proper reduct of M∼ and vice versa, and that the combinatorial pregeometries associated with both structures are isomorphic.