In type algebra, type isomorphism is the case when two types are interchangeable with no information lost.

That means, two types A and B are isomorphic to each other if any a: A can be casted or converted to B and back to the same a, and any b: B can be casted or converted to A and back to the same b.

For instance, take the following example:

sealed trait YesNo
object Yes extends YesNo
object No extends YesNo

The YesNo type is isomorphic to Boolean:

implicit def yesno2bool(value: YesNo): Boolean = value == Yes
implicit def bool2yesno(value: Boolean): YesNo = if (value) Yes else No

There’s no information lost in converting YesNo to Boolean and back.

Arity

The way to find whether two types are isomorphic is by their arities: if the arities are the same, the types are isomorphic.

Briefly, a type’s arity is the amount of its possible values. Consider a type as a set: the arity is the number of elements.

A few main examples:

• Nothing: arity is 0 (no possible instance)
• Unit: arity is 1 (only the () value)
• Boolean: arity is 2 (true and false)
• Int: arity is 4 294 967 296 (2³², 4 bytes)

So if A and B have the same arity, you can map each value in A to only one in B, and vice versa.

Algebraic Data Types

ADT (don’t confuse with ADT – abstract data type) are compounded types types under a compounding-operation viewpoint, and their arity follows simple arithmetic rules.

A disjunctive type – when the compounding values are exclusively optional, that means, one or another – has arity equal to the sum of its compounding types.

For instance, the YesNo type is the disjunction of Yes.type and No.type, it means a YesNo instance can be a Yes (instance of Yes.type) or a No (instance of No.type):

• Yes.type has arity 1
• No.type as arity 1
• YesNo ≡ Yes.type + No.type
• ⊢ The YesNo’s arity is 1 + 1 = 2

Note: Dotty (Scala 3) has a nice syntax for type disjunction, similar to Haskell’s.

A conjunctive type – when the compounding values are companions, that means, one and another – has arity equal to the product of its compounding types.

For instance, take the tuple (Boolean, Int):

• Boolean has arity 2
• Int has arity 4 294 967 296
• (Boolean, Int) ≡ Boolean * Int
• ⊢ The (Boolean, Int)’s arity is 2 × 4 294 967 296 = 8 589 934 592

A lambda type has arity equal to the return type’s arity raised to the power of the argument type’s – multiple arguments are equivalent to a conjunctive type.

• Boolean => Int has arity 4 294 967 296²
• Int => Boolean has arity 2⁴²⁹⁴⁹⁶⁷²⁹⁶

You can understand why reading this.

Swapping

Note that swappable types are isomorphic, cause a+b=b+a and ab=ba.

So A|B is isomorphic to B|A, and (A,B) is isomorphic to (B,A).

The proof is nothing trickier than the swap function itself: if you can swap the values without losing data, the types are isomorphic.

def swap[A, B](v: (A, B)): (B, A) = (v._2, v._1)

Curiosity: isomorphism in C

Since C has low-level access to the memory (through pointers) and is weakly typed, one can cast between the two most different types if they have the same memory length.

For instance, in C int and float are isomorphic.

And amusing code is the fast inverse square root form Quake Ⅲ Arena (Q_rsqrt), it does some type black magic with pointer casting:

float Q_rsqrt(float number) {
    long i;
    float x2, y;
    const float threehalfs = 1.5F;

    x2 = number * 0.5F;
    y  = number;
    i  = *(long *) &y;
    i  = 0x5f3759df - (i >> 1);
    y  = *(float *) &i;
    y  = y * (threehalfs - (x2 * y * y));

    return y;
}

In this example, long and float aren’t isomorphic, ’cause they have different arities, but it shows how data are easily interchangeable, promoting the isomorphism.

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