What are "vocabulary types", and how many exist?

Question

Across programming languages, I've encountered similar composite types with different names:

1. Optional / Maybe
2. Any
3. Variant / Sum
4. Record / Product

People often use the term vocabulary type,
yet I've never seen a definition of what makes a type "vocabulary".

Does this term have a loose definition?
What does type theory and other programming languages have to say about vocabulary types?

Is everything above a vocabulary type? Are there more?

Answer 1

My understanding of vocabulary with respect to programming and vocabulary type in general is something that gives to certain objects properties with well defined meaning.

Here are some examples in Haskell:

Consider Optional / Maybe type and a pure function that takes HTTP reply of a web server and extracts reply's code.

getCode :: String -> Int

Now suppose by the time we run this function we don't know if response was successful or not - code might not be present at all. How do we represent case when code is missing? We can assign some artificial value -1 or 0 or we can change whole function type to this:

getCode :: String -> Maybe Int

In addition Maybe forms a Monad, Functor, Applicative, Foldable and a bunch of other typeclasses in Haskell. Each typeclass adds extra abilities to manipulate value in question while respecting it's presence/absence.

Product / Sum type in Haskell are represented as pairs and Either a b. Again - defining something via Product or Sum adds a well defined meaning Product a b - both values must be present, Sum a b - one value must be present and adds a bunch of laws for free.