  Exploiting the interior-closure duality developed by Epstein and R.G., we
show that for the class of Matlis dualizable modules $\mathcal{M}$ over a
Noetherian local ring, when cl is a Nakayama closure and i its dual interior,
there is a duality between cl-reductions and i-expansions that leads to a
duality between the cl-core of modules in $\mathcal{M}$ and the i-hull of
modules in $\mathcal{M}^\vee$. We further show that many algebra and module
closures and interiors are Nakayama and describe a method to compute the
interior of ideals using closures and colons. We use our methods to give a
unified proof of the equivalence of F-rationality with F-regularity, and of
F-injectivity with F-purity, in the complete Gorenstein local case.
Additionally, we give a new characterization of the finitistic tight closure
test ideal in terms of maps from $R^{1/p^e}$. Moreover, we show that the
liftable integral spread of a module exists.
