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 non-empty 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

where is the usual multiplication of numbers (string, +) where + is the usual concatenation of strings

where is the usual concatenation of strings (boolean, &&) where && is the usual conjunction

and many more.

Type class definition

As usual in fp-ts the type class Semigroup , contained in the fp-ts/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 all x , y , z in A

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 non-empty 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 ' fp-ts/lib/Ord ' import { getMeetSemigroup , getJoinSemigroup } from ' fp-ts/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 fp-ts/lib/Semigroup module exports a getStructSemigroup combinator:



import { getStructSemigroup } from ' fp-ts/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 fp-ts , 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 ' fp-ts/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 ' fp-ts/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 ' fp-ts/lib/Semigroup ' import { getApplySemigroup , some , none } from ' fp-ts/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 ' fp-ts/lib/Semigroup ' import { getMonoid } from ' fp-ts/lib/Array ' import { ordNumber , contramap } from ' fp-ts/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