Perpendicularity in an Abelian Group

We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in and 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. 1. Introduction In [1, page 115], Graham et al. made the following suggestion: When , the integers and have no prime factors in common and we say that they are relatively prime. This concept is so important in practice, we ought to have a special notation for it; but alas, number theorists have not agreed on a very good one yet. Therefore we cry: HEAR US, O MATHEMATICIANS OF THE WORLD! LET US NOT WAIT ANY LONGER! WE CAN MAKE MANY FORMULAS CLEARER BY ADOPTING A NEW NOTATION NOW! LET US AGREE TO WRITE “ ”, AND TO SAY “ ？？IS PRIME TO？？ ,” IF？？ ？？AND？？ ？？ARE RELATIVELY PRIME. Like perpendicular lines do not have a common direction, perpendicular numbers do not have common factors. In fact, this cry had been answered even before it was made. Namely, in studying -groups (i.e., groups with a lattice structure), Birkhoff [2, page 295] defines that two positive elements and of an -group are disjoint if and uses the notation for disjoint elements. He also remarks that disjointness specializes to relative primeness in the -group of positive integers. A motivation for the present paper is to study how justified ultimately it is to use the symbol of perpendicularity to denote relative primeness. Does this practice rely only on the analogy between having no common direction and having no common factor or is there a deeper linkage to entitle this convention? This question leads us to ask which properties essentially establish the notion of perpendicularity in the algebraic context and what the most suitable algebraic context for the axiomatization of perpendicularity actually is; we have recently studied the axioms of perpendicularity from an elementary geometric point of view [3]. In an inner product space, perpendicularity obviously traces back to the inner product being zero. However, certain features of this perpendicularity can be shifted down to simpler algebraic structures. We will define perpendicularity in an Abelian group and examine it in Section 2. In Section 3, we will focus on perpendicularity in . Davis [4] defined perpendicularity in an Abelian group differently. In Section 4, we will introduce his approach and compare it with ours. Thereafter,