Since semigroups are such a fundamental abstraction of functional programming, this blog post will be longer than usual.
General definition
A semigroup is a pair (A, *)
in which A
is a nonempty set and *
is a binary associative operation on A
, i.e. a function that takes two elements of A
as input and returns an element of A
as output...
*: (x: A, y: A) => A
... while associative means that the equation
(x * y) * z = x * (y * z)
holds for all x
, y
, z
in A
.
Associativity simply tells us that we do not have to worry about parenthesizing an expression and can write x * y * z
.
Semigroups capture the essence of parallelizable operations
There are plenty of examples of semigroups:

(number, *)
where*
is the usual multiplication of numbers 
(string, +)
where+
is the usual concatenation of strings 
(boolean, &&)
where&&
is the usual conjunction
and many more.
Type class definition
As usual in fpts
the type class Semigroup
, contained in the fpts/lib/Semigroup
module, is implemented as a TypeScript interface
, where the operation *
is named concat
interface Semigroup<A> {
concat: (x: A, y: A) => A
}
The following law must hold

Associativity:
concat(concat(x, y), z) = concat(x, concat(y, z))
, for allx
,y
,z
inA
The name concat
makes a particular sense for arrays (see later) but, based on the context and the type A
for which we are implementing an instance, the semigroup operation can be interpreted with different meanings
 "concatenation"
 "merging"
 "fusion"
 "selection"
 "addition"
 "substitution"
and many more.
Instances
This is how we can implement the semigroup (number, *)
/** number `Semigroup` under multiplication */
const semigroupProduct: Semigroup<number> = {
concat: (x, y) => x * y
}
Note that you can define different semigroup instances for the same type. Here's the implementation of the semigroup (number, +)
where +
is the usual addition of numbers
/** number `Semigroup` under addition */
const semigroupSum: Semigroup<number> = {
concat: (x, y) => x + y
}
Another example, with strings this time
const semigroupString: Semigroup<string> = {
concat: (x, y) => x + y
}
I can't find an instance!
What if, given a type A
, you can't find an associative operation on A
? You can create a (trivial) semigroup instance for every type just using the following constructions
/** Always return the first argument */
function getFirstSemigroup<A = never>(): Semigroup<A> {
return { concat: (x, y) => x }
}
/** Always return the second argument */
function getLastSemigroup<A = never>(): Semigroup<A> {
return { concat: (x, y) => y }
}
Another technique is to define a semigroup instance for Array<A>
(*), called the free semigroup of A
.
function getArraySemigroup<A = never>(): Semigroup<Array<A>> {
return { concat: (x, y) => x.concat(y) }
}
and map the elements of A
to the singleton elements of Array<A>
function of<A>(a: A): Array<A> {
return [a]
}
(*) strictly speaking is a semigroup instance for non empty arrays of A
Note. Here concat
is the native array method, which kind of explains the initial choice for the name of the Semigroup
operation.
The free semigroup of A
is the semigroup whose elements are all possible nonempty finite sequences of elements of A
.
Deriving from Ord
There's another way to build a semigroup instance for a type A
: if we already have an Ord instance for A
, then we can "turn it" into a semigroup.
Actually two possible semigroups
import { ordNumber } from 'fpts/lib/Ord'
import { getMeetSemigroup, getJoinSemigroup } from 'fpts/lib/Semigroup'
/** Takes the minimum of two values */
const semigroupMin: Semigroup<number> = getMeetSemigroup(ordNumber)
/** Takes the maximum of two values */
const semigroupMax: Semigroup<number> = getJoinSemigroup(ordNumber)
semigroupMin.concat(2, 1) // 1
semigroupMax.concat(2, 1) // 2
Let's write some Semigroup
instances for more complex types
type Point = {
x: number
y: number
}
const semigroupPoint: Semigroup<Point> = {
concat: (p1, p2) => ({
x: semigroupSum.concat(p1.x, p2.x),
y: semigroupSum.concat(p1.y, p2.y)
})
}
This is mostly boilerplate though. The good news is that we can build a Semigroup
instance for a struct like Point
if we can provide a Semigroup
instance for each field.
Indeed the fpts/lib/Semigroup
module exports a getStructSemigroup
combinator:
import { getStructSemigroup } from 'fpts/lib/Semigroup'
const semigroupPoint: Semigroup<Point> = getStructSemigroup({
x: semigroupSum,
y: semigroupSum
})
We can go on and feed getStructSemigroup
with the instance just defined
type Vector = {
from: Point
to: Point
}
const semigroupVector: Semigroup<Vector> = getStructSemigroup({
from: semigroupPoint,
to: semigroupPoint
})
getStructSemigroup
is not the only combinator provided by fpts
, here's a combinator that allows to derive a Semigroup
instance for functions: given an instance of Semigroup
for S
we can derive an instance of Semigroup
for functions with signature (a: A) => S
, for all A
import { getFunctionSemigroup, Semigroup, semigroupAll } from 'fpts/lib/Semigroup'
/** `semigroupAll` is the boolean semigroup under conjunction */
const semigroupPredicate: Semigroup<(p: Point) => boolean> = getFunctionSemigroup(
semigroupAll
)<Point>()
Now we can "merge" two predicates on Point
s
const isPositiveX = (p: Point): boolean => p.x >= 0
const isPositiveY = (p: Point): boolean => p.y >= 0
const isPositiveXY = semigroupPredicate.concat(isPositiveX, isPositiveY)
isPositiveXY({ x: 1, y: 1 }) // true
isPositiveXY({ x: 1, y: 1 }) // false
isPositiveXY({ x: 1, y: 1 }) // false
isPositiveXY({ x: 1, y: 1 }) // false
Folding
By definition concat
works with only two elements of A
, what if we want to concat more elements?
The fold
function takes a semigroup instance, an initial value and an array of elements:
import { fold, semigroupSum, semigroupProduct } from 'fpts/lib/Semigroup'
const sum = fold(semigroupSum)
sum(0, [1, 2, 3, 4]) // 10
const product = fold(semigroupProduct)
product(1, [1, 2, 3, 4]) // 24
Semigroups for type constructors
What if we want to "merge" two Option<A>
? There are four cases:
x  y  concat(x, y) 

none  none  none 
some(a)  none  none 
none  some(a)  none 
some(a)  some(b)  ? 
There's a problem with the last one, we'd need something to "merge" two A
s.
That's what Semigroup
does! We can require a semigroup instance for A
and then derive a semigroup instance for Option<A>
. This is how the getApplySemigroup
combinator works
import { semigroupSum } from 'fpts/lib/Semigroup'
import { getApplySemigroup, some, none } from 'fpts/lib/Option'
const S = getApplySemigroup(semigroupSum)
S.concat(some(1), none) // none
S.concat(some(1), some(2)) // some(3)
Appendix
We've seen that semigroups help us any time we want to "concat", "merge", or "combine" (whatever word gives you the best intuition) several data into one.
Let's wrap all together with a final example (adapted from Fantas, Eel, and Specification 4: Semigroup)
Let's imagine you're building some system in which you store customer records that look like this:
interface Customer {
name: string
favouriteThings: Array<string>
registeredAt: number // since epoch
lastUpdatedAt: number // since epoch
hasMadePurchase: boolean
}
For whatever reason you might end up with duplicate records for the same person. What we need is a merge strategy. That's what semigroups are all about
import {
Semigroup,
getStructSemigroup,
getJoinSemigroup,
getMeetSemigroup,
semigroupAny
} from 'fpts/lib/Semigroup'
import { getMonoid } from 'fpts/lib/Array'
import { ordNumber, contramap } from 'fpts/lib/Ord'
const semigroupCustomer: Semigroup<Customer> = getStructSemigroup({
// keep the longer name
name: getJoinSemigroup(contramap((s: string) => s.length)(ordNumber)),
// accumulate things
favouriteThings: getMonoid<string>(), // <= getMonoid returns a Semigroup for `Array<string>` see later
// keep the least recent date
registeredAt: getMeetSemigroup(ordNumber),
// keep the most recent date
lastUpdatedAt: getJoinSemigroup(ordNumber),
// Boolean semigroup under disjunction
hasMadePurchase: semigroupAny
})
semigroupCustomer.concat(
{
name: 'Giulio',
favouriteThings: ['math', 'climbing'],
registeredAt: new Date(2018, 1, 20).getTime(),
lastUpdatedAt: new Date(2018, 2, 18).getTime(),
hasMadePurchase: false
},
{
name: 'Giulio Canti',
favouriteThings: ['functional programming'],
registeredAt: new Date(2018, 1, 22).getTime(),
lastUpdatedAt: new Date(2018, 2, 9).getTime(),
hasMadePurchase: true
}
)
/*
{ name: 'Giulio Canti',
favouriteThings: [ 'math', 'climbing', 'functional programming' ],
registeredAt: 1519081200000, // new Date(2018, 1, 20).getTime()
lastUpdatedAt: 1521327600000, // new Date(2018, 2, 18).getTime()
hasMadePurchase: true }
*/
The function getMonoid
returns a Semigroup
for Array<string>
. Actually it returns something more than a semigroup: a monoid.
So what's a monoid? In the next post I'll talk about Monoids
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Discussion
Hi Giulio, nice post.
I was just wondering why the semigroup for Option behaves the way you describe and not returning the non empty Option when combining an empty Option with a non empty one?
Thanks
That's one of the possible
Semigroup
instances, which corresponds to the "computation failed" meaning ofOption
(so if you combine two computations and one of them fails the combination as a whole fails too). There are other possible instances though, seegetFirstMonoid
,getLastMonoid
andgetMonoid
from the Option moduleI see, thanks for the clarification!