Metrics on doubles as an inverse semigroup II

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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 a description of the set E(M(X)) of idempotents of this inverse semigroup and of its Stone dual space Xˆ. We also construct σ-additive measures on Xˆ from finitely additive probability measures on X that vanish on bounded subsets.