The closure of a subset \(H\) of an inverse semigroup \(S\) consists of all those members of \(S\) which lie above or are equal to some member of \(H\) in the natural partial order. The poset of closed inverse subsemigroups of \(S\) forms a lattice and the central result is a four-way characterization of those inverse monoids \(S\) for which this lattice is distributive (all such \(S\) are cryptic) which includes their realization as those monoids which have the property that, for any \(a\in S\) and idempotent \(e\), there exists some \(a'\) in the inverse subsemigroup generated by \(a\) such that \(ea\geq a'e\). This result is then applied to semilattices of groups, \(E\)-unitary semigroups, and \(\omega\)-semigroups.