Metrics on doubles as an inverse semigroup II

TL;DR

This paper explores the algebraic structure of metrics on doubled spaces, characterizing idempotents and their dual spaces, and constructing measures linking the original space to its dual.

Contribution

It provides new descriptions of the idempotent set and Stone dual space of the inverse semigroup of metrics, extending previous work on coarse equivalence classes.

Findings

Descriptions of the idempotent set E(M(X))

Characterization of the Stone dual space E4X

Construction of -measures from finitely additive measures

Abstract

We have shown recently that, given a metric space $X$, the coarse equivalence classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$. Here we give several descriptions of the set $E(M(X))$ of idempotents of this inverse semigroup and its Stone dual space $\widehat X$. We also construct $\sigma$-additive measures on $\widehat X$ from finitely additive probability measures on $X$ that vanish on bounded subsets.