A Note On Perpendicularities And Inner Products

A perpendicularity perpendicular to in a module M is a binary relation that is irreflexive (but 0 perpendicular to 0), symmetric, serial, and preserves addition and scalar multiplication. If perpendicular to is not a subset of another perpendicularity in M, then perpendicular to is maximal. If M is a finite-dimensional vector space and perpendicular to is induced by an inner product, then it is well known that perpendicular to is maximal. We disprove the converse and an analogous result in the context of Abelian groups.