module Algebra.Semigroup where

# Semigroupsπ

record is-semigroup {A : Type β} (_β_ : A β A β A) : Type β where

A semigroup is an associative magma, that is, a set equipped with a choice of associative binary operation β.

field
has-is-magma : is-magma _β_
associative : {x y z : A} β x β (y β z) β‘ (x β y) β z

open is-magma has-is-magma public

open is-semigroup public

To see why the set truncation is really necessary, it helps to explicitly describe the expected structure of a ββ-semigroupβ in terms of the language of higher categories:

• An β-groupoid A, equipped with

• A map _β_ : A β A β A, such that

• β is associative: there exists an invertible 2-morphism Ξ± : A β (B β C) β‘ (A β B) β C (called the associator), satisfying

• The pentagon identity, i.e.Β there is a path Ο (called, no joke, the βpentagonatorβ) witnessing commutativity of the diagram below, where all the faces are Ξ±:

• The pentagonator satisfies its own coherence law, which looks like the Stasheff polytope and so on, βall the way up to infinityβ.

By explicitly asking that A be truncated at the level of sets, we have that the associator automatically satisfies the pentagon identity - because all parallel paths in a set are equal. Furthermore, by the upwards closure of h-levels, any further coherence condition you could dream up and write down for these morphisms is automatically satisfied.

As a consequence of this truncation, we get that being a semigroup operator is a property of the operator:

is-semigroup-is-prop : {_β_ : A β A β A} β is-prop (is-semigroup _β_)
is-semigroup-is-prop x y i .has-is-magma =
is-magma-is-prop (x .has-is-magma) (y .has-is-magma) i
is-semigroup-is-prop {_β_ = _β_} x y i .associative {a} {b} {c} =
x .has-is-set (a β (b β c)) ((a β b) β c) (x .associative) (y .associative) i

instance
H-Level-is-semigroup : β {_*_ : A β A β A} {n} β H-Level (is-semigroup _*_) (suc n)
H-Level-is-semigroup = prop-instance is-semigroup-is-prop

A semigroup structure on a type packages up the binary operation and the axiom in a way equivalent to a structure.

Semigroup-on : Type β β Type β
Semigroup-on X = Ξ£ (X β X β X) is-semigroup

Semigroup-on is a univalent structure, because it is equivalent to a structure expressed as a structure description. This is only the case because is-semigroup is a proposition, i.e.Β Semigroup-on can be expressed as a βstructure partβ (the binary operation) and an βaxiom partβ (the associativity)!

module _ where
private
sg-desc : Str-desc β β (Ξ» X β (X β X β X)) β
sg-desc .Str-desc.descriptor = sβ sβ (sβ sβ sβ)
sg-desc .Str-desc.axioms X = is-semigroup
sg-desc .Str-desc.axioms-prop X s = is-semigroup-is-prop

Semigroup-str : Structure β (Semigroup-on {β = β})
Semigroup-str = DescβStr sg-desc

Semigroup-str-is-univalent : is-univalent (Semigroup-str {β = β})
Semigroup-str-is-univalent = Descβis-univalent sg-desc

One can check that the notion of semigroup homomorphism generated by Semigroup-str corresponds exactly to the expected definition, and does not have any superfluous information:

module _
{A : Type} {_β_ : A β A β A} {as : is-semigroup _β_}
{B : Type} {_*_ : B β B β B} {bs : is-semigroup _*_}
{f : A β B}
where

_ : Semigroup-str .is-hom (A , _β_ , as) (B , _*_ , bs) f
β‘ ( (x y : A) β f .fst (x β y) β‘ (f .fst x) * (f .fst y))
_ = refl

## The βminβ semigroupπ

An example of a naturally-occurring semigroup are the natural numbers under taking minimums.

open import Data.Nat.Properties
open import Data.Nat.Order
open import Data.Nat.Base

Nat-min : is-semigroup min
Nat-min .has-is-magma .has-is-set = Nat-is-set
Nat-min .associative = min-assoc _ _ _

What is meant by βnaturally occurringβ is that this semigroup can not be made into a monoid: There is no natural number unit such that, for all y, min unit y β‘ y and min y unit β‘ y.

private
min-no-id : (unit : Nat) β Β¬ ((y : Nat) β min unit y β‘ y)
min-no-id x id =
let
sucxβ€x : suc x β€ x
sucxβ€x = subst (Ξ» e β e β€ x) (id (suc x)) (min-β€l x (suc x))
in Β¬sucxβ€x x sucxβ€x