Inverse semigroups of metrics on doubles related to certain subsets

TL;DR

This paper introduces a new inverse subsemigroup of metrics on doubles of a space, focusing on isometric subspaces like geodesic rays, with applications to Euclidean spaces and trees, enhancing computability.

Contribution

It defines a more computable inverse subsemigroup of metrics related to isometric subspaces, extending previous work on metric equivalence classes on doubles.

Findings

The subsemigroup is explicitly characterized for Euclidean spaces.

The subsemigroup is characterized for trees.

Applications include better understanding of metric structures related to geodesic rays.

Abstract

Recently we have shown that the equivalence classes of metrics on the double of a metric space $X$ form an inverse semigroup. Here we define an inverse subsemigroup related to a family of isometric subspaces of $X$, which is more computable. As a special case, we study this subsemigroup related to the family of geodesic rays starting from the basepoint, for Euclidean spaces and for trees.