  Relying on the notions of submodular function and partial metric, we
introduce normed inverse semigroups as a generalization of normed groups and
sup-semilattices equipped with an upper valuation. We define the property of
skew-convexity for a metric on an inverse semigroup, and prove that every norm
on a Clifford semigroup gives rise to a right-subinvariant and skew-convex
metric; it makes the semigroup into a Hausdorff topological inverse semigroup
if the norm is cyclically permutable. Conversely, we show that every Clifford
monoid equipped with a right-subinvariant and skew-convex metric admits a norm
for which the metric topology and the norm topology coincide. We characterize
convergence of nets and show that Cauchy completeness implies conditional
monotone completeness with respect to the natural partial order of the inverse
semigroup.
