This is the Algebras.Congruences module of the Agda Universal Algebra Library.

{-# OPTIONS --without-K --exact-split --safe #-} open import Algebras.Basic module Algebras.Congruences {𝑆 : Signature 𝓞 𝓥} where -- Imports from Agda and the Agda Standard Library ------------------------------ open import Agda.Primitive using ( _⊔_ ; lsuc ) renaming ( Set to Type ) open import Data.Product using ( Σ-syntax ; _,_ ) open import Function.Base using ( _∘_ ) open import Level using ( Level ) open import Relation.Binary using ( IsEquivalence ) renaming ( Rel to BinRel ) open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ) -- Imports from agda-algebras --------------------------------------------------- open import Overture.Preliminaries using ( ∣_∣ ; ∥_∥ ) open import Relations.Discrete using ( _|:_ ; 0[_] ) open import Relations.Quotients using ( 0[_]Equivalence ; _/_ ; ⟪_⟫ ; IsBlock ) open import Equality.Welldefined using ( swelldef ) open import Algebras.Products {𝑆 = 𝑆} using ( ov ) private variable α β ρ : Level

A *congruence relation* of an algebra `𝑨`

is defined to be an equivalence relation that is compatible with the basic operations of `𝑨`

. This concept can be represented in a number of alternative but equivalent ways.
Formally, we define a record type (`IsCongruence`

) to represent the property of being a congruence, and we define a Sigma type (`Con`

) to represent the type of congruences of a given algebra.

record IsCongruence (𝑨 : Algebra α 𝑆)(θ : BinRel ∣ 𝑨 ∣ ρ) : Type(ov ρ ⊔ α) where constructor mkcon field is-equivalence : IsEquivalence θ is-compatible : compatible 𝑨 θ Con : (𝑨 : Algebra α 𝑆) → Type(α ⊔ ov ρ) Con {α}{ρ}𝑨 = Σ[ θ ∈ ( BinRel ∣ 𝑨 ∣ ρ ) ] IsCongruence 𝑨 θ

Each of these types captures what it means to be a congruence and they are equivalent in the sense that each implies the other. One implication is the “uncurry” operation and the other is the second projection.

IsCongruence→Con : {𝑨 : Algebra α 𝑆}(θ : BinRel ∣ 𝑨 ∣ ρ) → IsCongruence 𝑨 θ → Con 𝑨 IsCongruence→Con θ p = θ , p Con→IsCongruence : {𝑨 : Algebra α 𝑆} → ((θ , _) : Con{α}{ρ} 𝑨) → IsCongruence 𝑨 θ Con→IsCongruence θ = ∥ θ ∥

We now defined the *zero relation* `0[_]`

and build the *trivial congruence*, which has `0[_]`

as its underlying relation. Observe that `0[_]`

is equivalent to the identity relation `≡`

and is obviously an equivalence relation.

open Level -- Example. The zero congruence of a structure. 0[_]Compatible : {α : Level}(𝑨 : Algebra α 𝑆){ρ : Level} → swelldef 𝓥 α → (𝑓 : ∣ 𝑆 ∣) → (𝑓 ̂ 𝑨) |: (0[ ∣ 𝑨 ∣ ]{ρ}) 0[ 𝑨 ]Compatible wd 𝑓 {i}{j} ptws0 = lift γ where γ : (𝑓 ̂ 𝑨) i ≡ (𝑓 ̂ 𝑨) j γ = wd (𝑓 ̂ 𝑨) i j (lower ∘ ptws0) open IsCongruence 0Con[_] : {α : Level}(𝑨 : Algebra α 𝑆){ρ : Level} → swelldef 𝓥 α → Con{α}{α ⊔ ρ} 𝑨 0Con[ 𝑨 ]{ρ} wd = let 0eq = 0[ ∣ 𝑨 ∣ ]Equivalence{ρ} in ∣ 0eq ∣ , mkcon ∥ 0eq ∥ (0[ 𝑨 ]Compatible wd)

A concrete example is `⟪𝟎⟫[ 𝑨 ╱ θ ]`

, presented in the next subsection.

In many areas of abstract mathematics the *quotient* of an algebra `𝑨`

with respect to a congruence relation `θ`

of `𝑨`

plays an important role. This quotient is typically denoted by `𝑨 / θ`

and Agda allows us to define and express quotients using this standard notation.

_╱_ : (𝑨 : Algebra α 𝑆) → Con{α}{ρ} 𝑨 → Algebra (α ⊔ lsuc ρ) 𝑆 𝑨 ╱ θ = (∣ 𝑨 ∣ / ∣ θ ∣) , -- the domain of the quotient algebra λ 𝑓 𝑎 → ⟪ (𝑓 ̂ 𝑨)(λ i → IsBlock.blk ∥ 𝑎 i ∥) ⟫ -- the basic operations of the quotient algebra

**Example**. If we adopt the notation `𝟎[ 𝑨 ╱ θ ]`

for the zero (or identity) relation on the quotient algebra `𝑨 ╱ θ`

, then we define the zero relation as follows.

𝟘[_╱_] : (𝑨 : Algebra α 𝑆)(θ : Con{α}{ρ} 𝑨) → BinRel (∣ 𝑨 ∣ / ∣ θ ∣)(α ⊔ lsuc ρ) 𝟘[ 𝑨 ╱ θ ] = λ u v → u ≡ v

From this we easily obtain the zero congruence of `𝑨 ╱ θ`

by applying the `Δ`

function defined above.

𝟎[_╱_] : {α : Level}(𝑨 : Algebra α 𝑆){ρ : Level}(θ : Con {α}{ρ}𝑨) → swelldef 𝓥 (α ⊔ lsuc ρ) → Con (𝑨 ╱ θ) 𝟎[_╱_] {α} 𝑨 {ρ} θ wd = let 0eq = 0[ ∣ 𝑨 ╱ θ ∣ ]Equivalence in ∣ 0eq ∣ , mkcon ∥ 0eq ∥ (0[ 𝑨 ╱ θ ]Compatible {ρ} wd)

Finally, the following elimination rule is sometimes useful (but it ‘cheats’ a lot by baking in a large amount of extensionality that is miraculously true).

open IsCongruence /-≡ : {𝑨 : Algebra α 𝑆}(θ : Con{α}{ρ} 𝑨){u v : ∣ 𝑨 ∣} → ⟪ u ⟫ {∣ θ ∣} ≡ ⟪ v ⟫ → ∣ θ ∣ u v /-≡ θ refl = IsEquivalence.refl (is-equivalence ∥ θ ∥)