Perpendicularity in an Abelian Group

Abstract

We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in (ℤn,+) and (ℚ+,·) and in certain other groups. Our approach provides a justification for the use of the symbol ⊥ denoting relative primeness in number theory and extends the domain of this convention to some degree. Related to that, we also consider parallelism from an axiomatic perspective.