Tuples of monoids form monoids. Data objects of monoids also form monoids. An article for object-oriented programmers.

This article is part of a series about monoids. In short, a monoid is an associative binary operation with a neutral element (also known as identity). This article starts off with some easy-to-understand, but abstract results. Once these are established, however, you'll see how to use them in a relatable example, so keep reading!


A tuple is a group of elements. In statically typed programming languages, each element has a type, and the types don't have to be the same. As an example, in C#, you can create a tuple like this:

Tuple<intstring> pair = Tuple.Create(42, "Foo");

This creates a tuple where the first element must be an int and the second element a string. In the example, I've explicitly declared the type instead of using the var keyword, but this is only to make the type clearer (since you don't have an IDE in which to read the code).

The pair tuple is a two-tuple, which means that it must have exactly two elements, of the types given, but you can also create larger tuples:

Tuple<stringboolint> triple = Tuple.Create("Bar"false, 42);

This is a three-tuple, but conceptually, tuples can have any size.

Pairs of monoids

A pair (a two-tuple) forms a monoid if both elements form a monoid. Haskell formalises this by stating:

instance (Monoid a, Monoid b) => Monoid (a, b)

The way to read this is that for any monoid a and any monoid b, the pair (a, b) is also a monoid.

Perhaps this is easiest to understand with a C# example. Consider a tuple of the type Tuple<int, string>. Integers form monoids under both addition and multiplication, and strings are monoids under concatenation. Thus, you can make Tuple<int, string> form a monoid as well. For instance, use the multiplication monoid to define this binary operation:

public static Tuple<intstring> CombinePair(
    Tuple<intstring> x,
    Tuple<intstring> y)
    return Tuple.Create(x.Item1 * y.Item1, x.Item2 + y.Item2);

For this particular example, I've chosen multiplication as the binary operation for int, and the string concatenation operator + for string. The point is that since both elements are monoids, you can use their respective binary operations to return a new tuple with the combined values.

This operation is associative, as the following FsCheck property demonstrates:

[Property(QuietOnSuccess = true)]
public void CombinePairIsAssociative(
    Tuple<intstring> x,
    Tuple<intstring> y,
    Tuple<intstring> z)
        CombinePair(CombinePair(x, y), z),
        CombinePair(x, CombinePair(y, z)));

This property passes for all the x, y, and z values that FsCheck generates.

The CombinePair operation has identity as well:

public static Tuple<intstring> PairIdentity = Tuple.Create(1, "");

Again, you can use the identity value for each of the elements in the tuple: 1 for the multiplication monoid, and "" for string concatenation.

This value behaves as the identity for CombinePair, at least for all non-null string values:

[Property(QuietOnSuccess = true)]
public void CombinePairHasIdentity(Tuple<intNonNull<string>> seed)
    var x = Tuple.Create(seed.Item1, seed.Item2.Get);
    Assert.Equal(CombinePair(PairIdentity, x), CombinePair(x, PairIdentity));
    Assert.Equal(x, CombinePair(x, PairIdentity));

Again, this test passes for all seed values generated by FsCheck.

The C# code here is only an example, but I hope it's clear how the result generalises.

Triples of monoids

In the above section, you saw how pairs of monoids form a monoid. Not surprisingly, triples of monoids also form monoids. Here's another C# example:

public static Tuple<stringboolint> CombineTriple(
    Tuple<stringboolint> x,
    Tuple<stringboolint> y)
    return Tuple.Create(
        x.Item1 + y.Item1,
        x.Item2 || y.Item2,
        x.Item3 * y.Item3);

The CombineTriple method is another binary operation. This time it combines two triples to a single triple. Since both string, bool, and int form monoids, it's possible to combine each element in the two tuples to create a new tuple. There's more than one monoid for integers, and the same goes for Boolean values, but here I've chosen multiplication and Boolean or, so the identity is this:

public static Tuple<stringboolint> TripleIdentity =
    Tuple.Create(""false, 1);

This triple simply contains the identities for string concatenation, Boolean or, and multiplication. The operation is associative, but I'm not going to show this with a property-based test. Both tests for associativity and identity are similar to the above tests; you could consider writing them as an exercise, if you'd like.

This triple example only demonstrates a particular triple, but you can find the generalisation in Haskell:

instance (Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)

This simply states that for monoids a, b, and c, the tuple (a, b, c) is also a monoid.


At this point, it can hardly come as a surprise that quadruples and pentuples of monoids are also monoids. From Haskell:

instance (Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)
instance (Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)

The Haskell standard library stops at pentuples (five-tuples), because it has to stop somewhere, but I'm sure you can see how this is a general rule.

Data objects as monoids

If you're an object-oriented programmer, you probably don't use tuples much in your day-to-day work. I'd even suggest that you shouldn't, because tuples carry too little information to make good domain objects. For example, if you have Tuple<int, string, string>, what do the elements mean? A better design would be to introduce a small Value Object called Customer, with Id, FirstName, and LastName properties.

(In functional programming, you frequently use tuples, because they're useful for 'gluing' generic functions together. A Haskell programmer may instead say that they are useful for composing parametrically polymorphic functions, but the meaning would be the same.)

As on object-oriented developer, then why should you care that tuples of monoids are monoids?

The reason this is interesting in object-oriented programming is that there's a strong relationship between tuples and data objects (Value Objects or Data Transfer Objects). Consider the Customer examples that I sketched out a few paragraphs above. As you'll learn in a future article, you can refactor a tuple to a class, or a class to a tuple.

Example: Roster

In Denmark, where I live, learning to swim is a mandatory part of the school curriculum. Teachers take the children to the nearby swimming stadium and teach them to swim. Since this is an activity outside of school, teachers would be prudent to keep a roster of the children. Modelled as a class, it might look like this:

public class Roster
    public int Girls { get; }
    public int Boys { get; }
    public IReadOnlyCollection<string> Exemptions { get; }
    public Roster(int girls, int boys, params string[] exemptions)
        Girls = girls;
        Boys = boys;
        Exemptions = exemptions;
    // ...

Some children may be temporarily exempt from a swimming lesson, perhaps because of a passing medical condition. This changes from lesson to lesson, so the roster keeps track of them separately. Additionally, the boys will need to go in the men's changing rooms, and the girls in the women's changing rooms. This is the reason the roster keeps track of the number of boys and girls separately.

This, however, presents a logistical problem, because there's only one teacher for a class. The children are small, so need the company of an adult.

The way my children's school solved that problem was to combine two groups of children (in Danish, en klasse, a class), each with their own teacher - one female, and one male.

To model that, the Roster class should have a Combine method:

public Roster Combine(Roster other)
    return new Roster(
        this.Girls + other.Girls,
        this.Boys + other.Boys,

Clearly, this is easy to implement. Just add the number of girls together, add the number of boys together, and concatenate the two lists of exemptions.

Here's an example of using the method:

public void UsageExample()
    var x = new Roster(11, 10, "Susan""George");
    var y = new Roster(12, 9, "Edward");
    var roster = x.Combine(y);
    var expected = 
        new Roster(23, 19, "Susan""George""Edward");
    Assert.Equal(expected, roster);

The Combine method is an instance method on the Roster class, taking a second Roster as input, and returning a new Roster value. It's a binary operation. Does it also have identity?

Yes, it does:

public static readonly Roster Identity = new Roster(0, 0);

Notice that the exemptions constructor argument is a params array, so omitting it means passing an empty array as the third argument.

The following properties demonstrate that the Combine operation is both associative and has identity:

[Property(QuietOnSuccess = true)]
public void CombineIsAssociative(Roster x, Roster y, Roster z)
[Property(QuietOnSuccess = true)]
public void CombineHasIdentity(Roster x)
    Assert.Equal(x, Roster.Identity.Combine(x));
    Assert.Equal(x, x.Combine(Roster.Identity));

In other words, Combine is a monoid.

This shouldn't surprise us in the least, since we've already established that tuples of monoids are monoids, and that a data class is more or less 'just' a tuple with named elements. Specifically, the Roster class is a 'tuple' of two addition monoids and the sequence concatenation monoid, so it follows that the Combine method is a monoid as well.

Roster isomorphism

In future articles, you'll learn more about isomorphisms between various representations of objects, but in this context, I think it's relevant to show how the Roster example is isomorphic to a tuple. It's trivial, really:

public Tuple<intintstring[]> ToTriple()
    return Tuple.Create(this.Girls, this.Boys, this.Exemptions.ToArray());
public static Roster FromTriple(Tuple<intintstring[]> triple)
    return new Roster(triple.Item1, triple.Item2, triple.Item3);

This pair of methods turn a Roster into a triple, and a corresponding triple back into a Roster value. As the following two FsCheck properties demonstrate, these methods form an isomorphism:

[Property(QuietOnSuccess = true)]
public void ToTripleRoundTrips(Roster x)
    var triple = x.ToTriple();
    Assert.Equal(x, Roster.FromTriple(triple));
[Property(QuietOnSuccess = true)]
public void FromTripleRoundTrips(Tuple<intintstring[]> triple)
    var roster = Roster.FromTriple(triple);
    Assert.Equal(triple, roster.ToTriple());

This isn't the only possible isomorphism between triples and Roster objects. You could create another one where the string[] element goes first, instead of last; where boys go before girls; and so on.


Tuples of monoids are also monoids. This holds for tuples of any size, but all of the elements has to be monoids. By isomorphism, this result also applies to data objects.

Next: Function monoids.


Hi Mark, I have trouble understanding your usage of the term 'monoid' in this post. You apply it to the types string, bool, and int when you say that a tuple of those "monoids" is a monoid as well. But up to this point you made it very clear, that a type is NOT a monoid. A function can be a monoid. So, would it be more correct to say that a tuple of certain functions, which are monoids, is a monoid as well?

2018-01-26 18:50 UTC

Punkislamist, thank you for writing. You're entirely correct that a monoid is an associative binary operation with identity. It's a function, not a type. If this article is unclear, the fault is all mine.

Not surprisingly, this topic is difficult to write about. The text has to be exact in order to avoid confusion, but since I'm only human, I sometimes make mistakes in how I phrase my explanations. While I've tried to favour the phrase that a type forms a monoid, I can see that I've slipped up once or twice in this article.

Some types form more than a single monoid. Boolean values, for instance, form exactly four monoids. Other types, like integers, form an infinite set of monoids, but the most commonly used integer monoids are addition and multiplication. Other types, particularly unit, only form a single monoid.

Why do I talk about types, then? There's at least two reasons. The first is the practical reason that most statically typed languages naturally come with a notion of types embedded. One could argue, I think, that types are a more fundamental concept than functions, since even functions have types (for instance, in Haskell, we'd characterise a binary operation with the type a -> a -> a).

A more abstract reason is that category theory mostly operates with the concepts of objects and morphisms. Such objects aren't objects in the sense of object-oriented programming, but rather correspond to types in programming languages. (Actually, a category theory object is a more fluffy concept than that, but that's the closest analogy that I'm aware of.)

In category theory, a particular monoid is an object in the category of monoids. For example, the integer addition monoid is an object in the category of monoids, as is the string concatenation monoid, etcetera.

When you consider a 'raw' programming language type like C#'s int, you're correct that it's not a monoid. It's just a type. The same goes for Haskell's corresponding Int32 type. As primitive values, we could say that the type of 32-bit integers is an object in some category (for example, the category of number representations). Such an object is not a monoid.

There exists, however, a morphism (a 'map') from the 32-bit integer object to the addition monoid (which is an object in the category of monoids). In Haskell, this morphism is the data constructor Sum:

Prelude Data.Monoid> :t Sum
Sum :: a -> Sum a

What this states is that Sum is a function (i.e. a morphism) that takes an object a and turns it into an object Sum a. We have to be careful here, because Sum a is a Haskell type, whereas Sum is the function that 'elevates' an object a to Sum a. The names are similar, but the roles are different. This is a common idiom in Haskell, and have some mnemonic advantages, but may be confusing until you get the hang of it.

We can think of Sum a as equivalent to the category theory object addition in the category of monoids. That's also how it works in Haskell: Sum a is a monoid:

Prelude Data.Monoid> Sum 40 <> Sum 2
Sum {getSum = 42}

In Haskell, <> is the polymorphic binary operation; exactly what it does depends on the object (that is: the type) on which it operates. When applied to two values of Sum a, the result of combining 40 and 2 is 42.

To be clear, Sum isn't the only morphism from the category of number representations to the category of monoids. Product is another:

Prelude Data.Monoid> :t Product
Product :: a -> Product a
Prelude Data.Monoid> Product 6 <> Product 7
Product {getProduct = 42}

Thus, there is a relationship between types and monoids, but it's most apparent in programming languages that are geared towards that way of thinking (like Haskell). In C#, it's difficult to translate some of these concepts into code, because C#'s type system isn't up to the task. Instead, when we consider a type like int, I think it's pragmatic to state that the type forms one or more monoids. I've also encountered the phrase that it gives rise to a monoid.

While you can represent a monoid with a C# interface, I've so far tried to avoid doing so, as I'm not sure whether or not it's helpful.

2018-01-28 11:57 UTC

Hi Mark, I did not expect to recieve such an exhaustive answer. That is incredible, thank you so much! It did clear up my confusion as well. Since most of these terms and concepts are new to me, even a slight inconsistency can be really confusing. But with your additional explanation I think I got a good understanding of the terms again.

Your explanations of these concepts in general are very well written and make it easy for people unfamiliar with this topic to understand the terms and their significance. Thanks again for writing!

2018-01-28 12:47 UTC

Punkislamist, thank you for those kind words. I'm happy to hear that what I wrote made sense to you; it makes sense to me, but I forgot to point out that I'm hardly an expert in category theory. Writing out the above answer helped clarify some things for me as well; as is common wisdom: you only really understand a topic when you teach it.

2018-01-28 20:59 UTC

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Monday, 30 October 2017 07:01:00 UTC


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