What's That Typeclass: Monoid
In programming, there’s a pattern that occurs very frequently – putting together two things of the same type to get another thing of that type.
a > a > a
Given its frequency, it would be nice to have some useful abstractions for it.
And in Haskell, we do. There’s a typeclass called Monoid
that abstracts over the notion of “smashing things together”. It draws inspiration from a mathematical structure with the same name.
In this article, we’ll cover both the typeclass and the structure.
By the end of the article, you will know:
 what is a monoid;
 what is the
Monoid
typeclass in Haskell;  how to use predefined monoid instances from
Data.Monoid
;  how to define your own instances of
Monoid
;  why are monoids useful.
Recommended previous knowledge: algebraic data types and typeclasses.
This article is also available in video form on our YouTube channel.
Building up intuition
With math terms like monoid, it’s always better to look at examples before reading the definition.
Therefore, let’s look at three examples of monoidlike behaviors in Haskell.
First off, lists with the ++
(concatenation) operator.
Prelude> [1,2] ++ [3, 4]  You can concatenate lists.
[1,2,3,4]
Prelude> [1,2] ++ []  Concatenating an empty list to a list doesn't change the result.
[1,2]
Prelude> [] ++ [1,2]
[1,2]
Prelude> [1,2] ++ ([3,4] ++ [5,6])  It doesn't matter in which order you concatenate the lists, you get the same result either way.
[1,2,3,4,5,6]
Prelude> ([1,2] ++ [3,4]) ++ [5,6]
[1,2,3,4,5,6]
After that, numbers with the +
operator.
Prelude> 1 + 2  You can sum two natural numbers together.
3
Prelude> 1 + 0  Adding a 0 doesn't change the sum.
1
Prelude> 0 + 1
1
Prelude> (1 + 2) + 3  It doesn't matter in which order you sum the numbers, you get the same result either way.
6
Prelude> 1 + (2 + 3)
6
(Alternatively, we could have also done numbers with the * operator. In that case, the element that doesn’t change the product would be 1.)
And, finally, booleans with the &&
operator (which stands for and
).
Prelude> True && False  You can join two booleans via &&.
False
Prelude> a = False
Prelude> a && True  Adding && True doesn't impact the resulting boolean.
False
Prelude> True && a
False
Prelude> (True && True) && False  It doesn't matter in which order you resolve the &&s, you get the same result either way.
False
Prelude> True && (True && False)
False
(Alternatively, we could have done booleans with the or operator: 
. In that case, the element that wouldn’t change the result would be False
.)
As you can see, these three things act similarly. They follow the same laws.
Now, let’s look at what exactly these laws are.
What is a monoid?
In mathematics, a monoid is a structure that consists of a set of elements (such as numbers or booleans) and a binary operation on that set (such as +
or &&
).
In addition, a monoid satisfies the following properties:
 There is an identity element that “doesn’t do anything”. In more formal terms, if you take any element x from the monoid’s element set and use the binary operation of the monoid with that element and the identity element, you will get the same element – x – back. E.g., $1 + 0 = 1$, $2 + 0 = 2$, etc.
 Associativity. This property guarantees that rearranging the parentheses in the equation won’t change the result of the equation. E.g., $(1 + 2) + 3 = 1 + (2 + 3)$.
Monoid
typeclass in Haskell
Monoid
typeclass in HaskellHow does a monoid look in Haskell? :info
detective is on the case. 🕵️♂️
Prelude> :info Monoid
type Monoid :: * > Constraint
class Semigroup a => Monoid a where
mempty :: a
mappend :: a > a > a
mconcat :: [a] > a
As we can see, Monoid
in Haskell is a typeclass that has three methods: mempty
, mappend
, and mconcat
.
mempty
is a value that doesn’t impact the result when used together with the binary operation. In other words, it’s the identity element of the monoid.mappend
(or<>
) is a function that puts two monoids together. In other words, it’s the binary operation of the monoid.mconcat
is a function that reduces a list of monoids into one value. By default, it’sfoldr mappend mempty
. That’s fine for most data types, and it’s not necessary to definemconcat
to define an instance. But you might sometimes want to define your own implementation of the function when the default implementation is not optimal.
Note on mappend
and <>
.
While the typeclass doesn’t define a function called <>
, it’s defined by Semigroup
, the superclass of Monoid
that I’ll cover later in the article. For all intents and purposes, these functions should be the same. In a future release of GHC, mappend
will be removed, so you are advised to use <>
.
How to use predefined monoid instances
Let’s try to use these monoid methods with data types from our previous examples.
It works with lists.
Prelude> [1,2] <> [3,4]
[1,2,3,4]
Prelude> mempty :: [a]
[]
Prelude> [1,2] <> mempty
[1,2]
But if you try to do 1 <> 3
, you will be greeted by an error in GHCi.
<interactive>:1:1: error:
• Ambiguous type variable 'a0' arising from a use of 'print'
...
🤔
This happens because there’s no single instance of Monoid
for numbers.
You can have a sum monoid, a product monoid, and many more, depending on the binary operation you choose. GHC has no way of knowing which operation you want to use.
So we need to wrap our data types and make monoid instances according to the context those types will be used in. Data.Monoid
defines such wrapper types for commonly used monoids like Sum
, Product
, etc.
newtype Sum a = Sum { getSum :: a }
Let’s see how they work.
Prelude> import Data.Monoid
Prelude Data.Monoid> Sum 1 <> Sum 3
Sum {getSum = 4}
Prelude Data.Monoid> Product 2 <> Product 5
Product {getProduct = 10}
Similarly, booleans have All
and Any
monoids defined for them.
Prelude Data.Monoid> All True <> All False
All {getAll = False}
Prelude Data.Monoid> Any True <> Any False
Any {getAny = True}
You can use mconcat
to sum a list of these monoids.
result = mconcat [Sum 4, Sum 6, Sum 8, mempty]
And to get an unwrapped result, you can unwrap them via their record names, conveniently called getX
.
Prelude Data.Monoid> getSum result
18
How to create a monoid instance in Haskell
Let’s try creating our own monoid instance in Haskell.
First, we’ll create a custom datatype called Move
that expresses instructions for a robot to move in a 2D field.
data Move = Move Int Int deriving (Show, Eq)
To create a monoid instance for a data type, you first need to create a Semigroup
instance for it because Semigroup
is a superclass of Monoid
(since GHC 8.4).
Prelude> :info Monoid
type Monoid :: * > Constraint
class Semigroup a => Monoid a where
...
Prelude> :info Semigroup
type Semigroup :: * > Constraint
class Semigroup a where
(<>) :: a > a > a
But don’t worry, there’s nothing new here. Creating a Semigroup
instance is just a roundabout way of defining mappend
.
This is so because a semigroup is a monoid without an identity element ( mempty
). It defines one method – <>
– which is the same as mappend
, the binary operation for combining two values.
So, let’s define an instance of Semigroup
. To append two moves, we will sum their respective x
and y
values.
instance Semigroup Move where
Move x1 y1 <> Move x2 y2 = Move (x1 + x2) (y1 + y2)
After that, we can define the Monoid
instance. To define the instance, you only need to provide mempty
because mappend
will be the same as <>
.
The mempty
that makes sense in our case is to not move anywhere: Move 0 0
.
instance Monoid Move where
mempty = Move 0 0
Note: while creating your monoid instances, you need to be careful to follow the monoid laws. Haskell (barring propertybased tests) has no way to check that our monoid instance makes sense. For example, we could have defined mempty
to be Move 0 (1)
, which would lead to some odd behavior.
That’s all we need for Move
to be a monoid! 🎉
Now, we can play around with it in GHCi.
*Main> Move 3 4 <> Move 8 9
Move 11 13
*Main> Move 3 4 <> mempty
Move 3 4
We can also use the mconcat
method to fold a list of moves.
*Main> stay = mempty :: Move
*Main> moves = [Move 1 2, Move (3) 5, Move (6) 3, stay, Move 2 3]
*Main> mconcat moves
Move (6) 13
Why do you need monoids?
In general, monoids are not very advanced, interesting, or cool.
However, they are really useful and quite simple. <>
is used for concatenating a lot of different types and could be considered a basic building block. It’s an operator that you will meet in every Haskell project.
If you’re searching for reallife use cases, the further learning section below has resources that cover the use of monoids for making a chat filter and calculating electoral vote distributions.
Further learning
This has been a quick introduction to the Monoid
typeclass in Haskell. Hopefully, now you know what monoid laws are, how to use monoids in Haskell, and how to define your own instances of the Monoid
typeclass.
Here’s some other resources on the subject that you might find helpful:
 How Monoids are useful in Programming? I absolutely love Tsoding’s videos about Haskell. In this one, he shows how monoids can be useful while implementing a filter for chat messages.
 Electoral vote distributions are monoids. Here’s another example of how monoids can be used to solve a realworld problem.
 Monoids in Haskell. Best introductory article I could find on the topic.
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Exercises

Implement a wrapper type for exclusive OR (XOR). Create Semigroup and Monoid instances for it.
Solution
newtype Xor = Xor {getXor :: Bool} deriving (Show, Eq) instance Semigroup Xor where Xor bool1 <> Xor bool2 = Xor (bool1 /= bool2) instance Monoid Xor where mempty = Xor False

Implement a wrapper type for logical biconditional (XNOR). Create Semigroup and Monoid instances for it.
Solution
newtype Xnor = Xnor {getXnor :: Bool} deriving (Show, Eq) instance Semigroup Xnor where Xnor bool1 <> Xnor bool2 = Xnor (bool1 == bool2) instance Monoid Xnor where mempty = Xnor True

Do the instances created in 1. and 2. follow the monoid laws? Why?
Solution
Answer: There are two laws that monoids need to satisfy: the existence of an identity element and associativity.
It’s easy to determine that False and True are the identity elements of XOR and XNOR, respectively, by looking at the truth tables of the operations.
Proving associativity is a bit harder, but here’s an intuitive proof on Stack Exchange for XOR. The one for XNOR should be equivalent in structure.

As you might remember, a semigroup is a monoid without the identity element. Can you come up with an example of a structure that is a semigroup but is not a monoid?
Solution
One example is nonempty lists with the
++
operator. While a regular list can be empty, and the empty list thus satisfies the requirement of an identity element, nonempty lists have no such element.
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