Inverse semigroups of metrics on doubles related to certain subsets (Q6174387)

In his earlier paper [J. Geom. Anal. 31, No. 6, 5721--5739 (2021; Zbl 1509.20115)], the author showed that for a given metric space \(X\), the set \(M(X)\) of coarse equivalence classes of metric spaces on the double \(X\sqcup X\) has the structure of an inverse semigroup. In the paper under review, he points out that this \(M(X)\) is too great to be computed by showing that for \(X = [0, \infty)\) with the standard metric, the cardinality of \(M(X)\) is at least continuum. The purpose of the paper is to define an inverse subsemigroup in \(M(X)\) that is substantially smaller and more computable than \(M(X)\). The idea for the construction of such an inverse subsemigroup is to find a family \(\Phi\) of isometric subsets and to define an inverse subsemigroup \(M(X)_{\Phi}\) and a semigroup homomorphism of \(M(X)_{\Phi}\) onto the semigroup \(PB(\Phi)\) of partial bijections of the family \(\Phi\). As a special case, he studies the subsemigroup related to the family of geodesic rays starting from the basepoint, for Euclidean spaces and for trees.