Gödel algebras: interactive dualities and their applications

We present a technique for deriving certain new natural dualities for any variety of algebras generated by a finite Heyting chain. The dualities we construct are tailored to admit a transparent translation to the more pictorial Priestley/Esakia duality and back again. This enables us to combine the two approaches and so to capitalise on the virtues of both, in particular the categorical good behaviour of a natural duality: we thereby demonstrate the fullness, or not, of each of our dualities; we obtain new results on amalgamation; and we also provide a simple treatment of coproducts.