What type of math is this: a -> b -> c

Question

I often see type declarations similar to this when looking at Haskell:

a -> (b -> c)

I understand that it describes a function that takes in something of type a and returns a new function that takes in something of type b and returns something of type c. I also understand that types are associative (edit: I was wrong about this - see the comments below), so the above could be rewritten like this to get the same result:

(a -> b) -> c

This would describe a function that takes in something of type a and something of type b and returns something of type c.

I've also heard that you can make a complement (edit: really, the word I was looking for here is dual - see the comments below) to the function by switching the arrows:

a <- b <- c

which I think is equivalent to

c -> b -> a

but I'm not sure.

My question is, what is the name of this kind of math? I'd like to learn more about it so I can use it to help me write better programs. I'm interested in learning things like what a complimentary function is, and what other transformations can be performed on type declarations.

Thanks!

Answer 1

Type declarations are not associative, a -> (b -> c) is not equivalent to (a -> b) -> c. Also, you can't "switch" the arrows, a <- b <- c is not valid syntax.

The usual reference to associativity is in this case that -> it right associative, which means that a -> b -> c is interpreted as a -> (b -> c).