Finitely represented /total/ maps
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The usual finite map type (Map from Data.Map) is not an applicative functor, as it doesn't have a pure. Consequently, it's also not a monad. On the other hand, we can decompose Map into two parts: a total map, and Maybe, i.e.,

type Map k v = TMap k (Maybe v)

The type TMap of total maps does have Applicative and Monad instances, and hence this hypothetically rebuilt Map would as well.

The idea for TMap is introduced in the paper Denotational design with type class morphisms. The meaning Map k v is given by its semantic function

(!) :: Map k v -> (k -> v)

The type class morphism (TCM) principle then exactly dictates the meanings of several class instances for TMap, including Functor, Applicative, Monad, and Monoid. For instance, (!) must be a monoid (homo)morphism, i.e.,

(!) mempty == mempty
(!) (s `mappend` t) == (!) s `mappend` (!) t

The current implementation of TMap is via Data.Map.