  Extensions of Stone-type dualities have a long history in algebraic logic and
have also been instrumental for proving results in algebraic language theory.
We show how to extend abstract categorical dualities via monoidal adjunctions,
subsuming various incarnations of classical extended Stone and Priestley
duality as a special case. Guided by these categorical foundations, we
investigate residuation algebras, which are algebraic models of language
derivatives, and show the subcategory of derivation algebras to be dually
equivalent to the category of profinite ordered monoids, restricting to a
duality between boolean residuation algebras and profinite monoids. We further
extend this duality to capture relational morphisms of profinite ordered
monoids, which dualize to natural morphisms of residuation algebras.
