You remember that Functional Programming (FP) is all about function composition? But what’s composition by the way? Let’s explain it here with a few other useful concepts for FP, and go one step further in monads understanding.

**Integer composition**

A composition of integer in mathematics is a way of writing *N* as the sum of a sequence strictly positive. Weak composition is for sequence positive or equal to zero. It means that every positive integer admits infinitely many weak compositions. 0 is also known as the neutral element for the set of integers.

**Function composition**

A function composition is when we apply a function (say *f) *to the result of another one (say *g*) to produce a third function (say *h).* We write it *h=g°f. *Intuitively, composing two functions is a chaining process in which the output of the inner function becomes the input of the outer function. This is in essence functional programming.

**Identity Function**

A function that returns exactly its input as the output without any alteration is also known as the identity function. There is a symetry between weak composition of integers and weak composition of functions, because an identity function is a neutral element for the set of functions. Like 0 for integers, it can help in several situations to allow more composition. Which is useful, because we want to compose everything!

**Which functions are composable?**

Two functions are composable if their Domain and Co-Domain are compatible. Let’s imagine two functions:

**StringToInt : s (string) -> i (integer)**StringToInt is a function mapping a string (Domain) with an integer (Co-Domain).**TwoTimes: i (integer) -> o (integer)**TwoTimes is a function mapping an integer (Domain) with another integer (Co-Domain).

Can I imagine a composition **TwoTimes>>StringToInt**? No because **TwoTimes **output is an integer, and **StringToInt** input is a string. The Co-Domain of **TwoTimes is **not compatible with the Domain of **StringToInt**. But the Co-Domain of **StringToInt **is compatible with the Domain of **TwoTimes**. Thus, we could define a correct composition **StringToInt>> TwoTimes**. Most FP patterns are just a way to wrap data so that Domain and Co-Domain stay compatible, thus composable, despite side effects.

**Composable functions properties**

If functions are composable, they can be associative if the Domain and the Co-Domain are suitable. It means: *g°f(x) = g(x)°f* .

You know for example that addition is associative.

So if *g* is *+5*, *f* is *+10 *and if *x is 3,* we can write: 5+ (3+10) = (5+3) + 10 = 18.

Note that, if two functions have the same Domain and Co-Domain, they are composable, and associative. The composition of functions with the same Domain and Co-Domain are called **transformations**, because mathematicians like to give a name to everything (and for once it seems a logical name).

**How powerful is function composition?**

In a few words: it allows to express complex things in a simple form. In physics for example, Lorentz Transformations are very famous. Using a series of transformation, Lorentz showed that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of Einstein’s Special Relativity theory.

Of course we’re not Einstein, but expressing complex things in a simple form is definitely something interesting for software design.

I think there might be a mistake (or at least the way you write it is confusing) about function composition.

First you write:

> “Can I imagine a composition TwoTimes>>StringToInt? No[…]”

> “[…] we could define a correct composition StringToInt>> TwoTimes.”

Then, several lines after you write:

> “If functions are composable, they are associative.”

From my understanding, composition is not defined (or constrained) by associativity, then I am confused with your latest sentence.

Hi there, thanks for reading and for the question.

If I believe Wikipedia: https://en.wikipedia.org/wiki/Function_composition

“The composition of functions is always associative—a property inherited from the composition of relations […] with suitably chosen domain and co domain”

“In a strict sense, the composition g ∘ f can be built only if f’s codomain equals g’s domain; in a wider sense it is sufficient that the former is a subset of the latter”

Thus I agree that my example is misleading, because we need suitably chosen domain and co domain to guarantee composition. I’ll try to clarify it.