Intrinsic generalized metrics

Abstract

The intrinsic functions of two variables from a lattice-ordered group to itself that are symmetric and right invariant are called its intrinsic metrics. It is known that these are exactly the functions of the form d(x, y) = n|x - y| for some integer n, and that for ${n \geq 1}$ , the triangle inequality for these functions holds if and only if the group is abelian.

Quasimetrics and, more recently, partial metrics (introduced for computer science applications), can be used to express some topologies on ordered sets (such as the usual topology on ${\mathbb{R}}$) as the join of two subtopologies whose open sets are respectively, upper and lower sets in the order. Thus, it is natural to look at intrinsic “partial metrics” and “quasimetrics” on lattice-ordered groups. Here, we define and characterize these intrinsic generalized metrics and obtain results relating commutativity to their key properties such as the triangle inequality.