# # Type algebra

## # Addition and multiplication

The addition and multiplication have equivalents in this type algebra. They correspond to the **tagged unions** and **product types**.

```
data Sum a b = A a | B b
data Prod a b = Prod a b
```

We can see how the number of inhabitants of every type corresponds to the operations of the algebra.

Equivalently, we can use `Either`

and `(,)`

as type constructors for the addition and the multiplication. They are isomorphic to our previously defined types:

```
type Sum' a b = Either a b
type Prod' a b = (a,b)
```

The expected results of addition and multiplication are followed by the type algebra up to isomorphism. For example, we can see an isomorphism between 1 + 2, 2 + 1 and 3; as 1 + 2 = 3 = 2 + 1.

```
data Color = Red | Green | Blue
f :: Sum () Bool -> Color
f (Left ()) = Red
f (Right True) = Green
f (Right False) = Blue
g :: Color -> Sum () Bool
g Red = Left ()
g Green = Right True
g Blue = Right False
f' :: Sum Bool () -> Color
f' (Right ()) = Red
f' (Left True) = Green
f' (Left False) = Blue
g' :: Color -> Sum Bool ()
g' Red = Right ()
g' Green = Left True
g' Blue = Left False
```

### # Rules of addition and multiplication

The common rules of commutativity, associativity and distributivity are valid because there are trivial isomorphisms between the following types:

```
-- Commutativity
Sum a b <=> Sum b a
Prod a b <=> Prod b a
-- Associativity
Sum (Sum a b) c <=> Sum a (Sum b c)
Prod (Prod a b) c <=> Prod a (Prod b c)
-- Distributivity
Prod a (Sum b c) <=> Sum (Prod a b) (Prod a c)
```

## # Functions

Functions can be seen as exponentials in our algebra. As we can see, if we take a type `a`

with n instances and a type `b`

with m instances, the type `a -> b`

will have m to the power of n instances.

As an example, `Bool -> Bool`

is isomorphic to `(Bool,Bool)`

, as 2*2 = 2².

```
iso1 :: (Bool -> Bool) -> (Bool,Bool)
iso1 f = (f True,f False)
iso2 :: (Bool,Bool) -> (Bool -> Bool)
iso2 (x,y) = (\p -> if p then x else y)
```

## # Natural numbers in type algebra

We can draw a connection between the Haskell types and the natural numbers. This connection can be made assigning to every type the number of inhabitants it has.

### # Finite union types

For finite types, it suffices to see that we can assign a natural type to every number, based in the number of constructors. For example:

```
type Color = Red | Yellow | Green
```

would be **3**. And the `Bool`

type would be **2**.

```
type Bool = True | False
```

### # Uniqueness up to isomorphism

We have seen that multiple types would correspond to a single number, but in this case, they would be isomorphic. This is to say that there would be a pair of morphisms `f`

and `g`

, whose composition would be the identity, connecting the two types.

```
f :: a -> b
g :: b -> a
f . g == id == g . f
```

In this case, we would say that the types are **isomorphic**. We will consider two types equal in our algebra as long as they are isomorphic.

For example, two different representations of the number two are trivally isomorphic:

```
type Bit = I | O
type Bool = True | False
bitValue :: Bit -> Bool
bitValue I = True
bitValue O = False
booleanBit :: Bool -> Bit
booleanBit True = I
booleanBit False = O
```

Because we can see `bitValue . booleanBit == id == booleanBit . bitValue`

### # One and Zero

The representation of the number **1** is obviously a type with only one constructor. In Haskell, this type is canonically the type `()`

, called Unit. Every other type with only one constructor is isomorphic to `()`

.

And our representation of **0** will be a type without constructors. This is the **Void** type in Haskell, as defined in `Data.Void`

. This would be equivalent to a unhabited type, wihtout data constructors:

```
data Void
```

## # Recursive types

### # Lists

Lists can be defined as:

```
data List a = Nil | Cons a (List a)
```

If we translate this into our type algebra, we get

List(a) = 1 + a * List(a)

But we can now substitute **List(a)** again in this expression multiple times, in order to get:

List(a) = 1 + a + a*a + a*a*a + a*a*a*a + ...

This makes sense if we see a list as a type that can contain only one value, as in `[]`

; or every value of type `a`

, as in `[x]`

; or two values of type `a`

, as in `[x,y]`

; and so on. The theoretical definition of List that we should get from there would be:

```
-- Not working Haskell code!
data List a = Nil
| One a
| Two a a
| Three a a a
...
```

### # Trees

We can do the same thing with binary trees, for example. If we define them as:

```
data Tree a = Empty | Node a (Tree a) (Tree a)
```

We get the expression:

Tree(a) = 1 + a * Tree(a) * Tree(a)

And if we make the same substitutions again and again, we would obtain the following sequence:

Tree(a) = 1 + a + 2 (a*a) + 5 (a*a*a) + 14 (a*a*a*a) + ...

The coefficients we get here correspond to the Catalan numbers sequence, and the n-th catalan number is precisely the number of possible binary trees with n nodes.

## # Derivatives

The derivative of a type is the type of its type of one-hole contexts. This is the type that we would get if we make a type variable disappear in every possible point and sum the results.

As an example, we can take the triple type `(a,a,a)`

, and derive it, obtaining

```
data OneHoleContextsOfTriple = (a,a,()) | (a,(),a) | ((),a,a)
```

This is coherent with our usual definition of derivation, as:

d/da (a*a*a) = 3*a*a

More on this topic can be read on this article.