Four relations on the set of point-hyperplane anti-flags

Abstract

There are precisely four arrangements of two point-hyperplane anti-flags. We consider the corresponding relations on the set of such anti-flags and show that each of them can be recovered from any other except in one special case. If the field consists of two elements, then one of the relations cannot be used to recover each of the remaining three. This is related to a bijection between anti-flags and exterior points of the hyperbolic polar space which exists in this case.