Why “Algebraic data type” use “Algebraic” in the name?

Question

When I learn Scala/Haskell, I see there is a concept of Algebraic data type. I\'ve read the explanation from the wikipedia, but I still have a question:

Why does it

Answer 1

Consider the type Bool. This type, of course, can take on one of two possible values: True or False.

Now consider

data EitherBool = Left Bool | Right Bool

How many values can this type take on? There are 4: Left False, Left True, Right False, Right True. How about

data EitherBoolInt = Left Bool | Right Int8

Here there are 2 possible values in the Left branch, and 2^8 in the Right branch. For a total of 2 + 2^8 possible values for EitherBoolInt. It should be easy to see that for any set of constructors and types, this kind of construction will give you a datatype with a space of possible values the size of the sum of the possible values of each individual constructor. For this reason, it's called a sum type.

Consider instead

data BoolAndInt = BAndI Bool Int8

or simply

type BoolAndInt = (Bool, Int)

How many values can this take on? For each possible Int8, there are two BoolAndInts, for a total of 2*2^8 = 2^9 total values. The total number of possible values is the product of the number of values of each field of the constructor, so this is called a product type.

This idea can be extended further -- for example, functions from a->b are an exponential datatype (see The Algebra of Algebraic Datatypes). You can even create a reasonable notion of the derivative of a datatype. This is not even a purely theoretical idea -- it's the basis for the functional construct of "zippers". See The Derivative of a Datatype is the Type of its One-Hole Contexts and The Wikipedia entry on zippers.