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Given any quasi-countable, in particular any countable inverse semigroup $S$,we introduce a way to equip $S$ with a proper and right subinvariant extendedmetric. This generalizes the notion of proper, right invariant metrics fordiscrete countable groups. Such a metric is shown to be unique up to bijectivecoarse equivalence of the semigroup, and hence depends essentially only on $S$.This allows us to unambiguously define the uniform Roe algebra of $S$, which weprove can be realized as a canonical crossed product of $\ell^\infty(S)$ and$S$. We relate these metrics to the analogous metrics on Hausdorff \'{e}talegroupoids.  Using this setting, we study those inverse semigroups with asymptoticdimension $0$. Generalizing results known for groups, we show that these areprecisely the locally finite inverse semigroups, and are further characterizedby having strongly quasi-diagonal uniform Roe algebras. We show that, unlike inthe group case, having a finite uniform Roe algebra is strictly weaker and ischaracterized by $S$ being locally $\mathcal L$-finite, and equivalently sparseas a metric space.

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups