Classical.Theories.Monoid¶

The equational theory of monoids¶

This is the Classical.Theories.Monoid module of the Agda Universal Algebra Library.

Th-Monoid has three equations: associativity, left identity, and right identity, composed from the generic builders of Classical.Equations applied to Sig-Monoid's symbols. Associativity needs three variables, the identity laws one each, so the variable carrier is uniformly Fin 3 (per ADR-002 v2 §2); the identity equations use 0F and ignore 1F, 2F. The codomain is spelled in long form, not _, per §4. This module's prose carries the same normative weight for the identity-bearing structures as Classical.Theories.Semigroup does for the associativity-only ones.

{-# OPTIONS --cubical-compatible --exact-split --safe #-}

module Classical.Theories.Monoid where

-- Imports from Agda and the Agda Standard Library ----------------------------
open import Agda.Primitive                         using () renaming ( Set to Type )
open import Data.Fin.Base                          using ( Fin )
open import Data.Fin.Patterns                      using ( 0F ; 1F ; 2F )
open import Data.Product                           using ( _×_ )
open import Relation.Binary.PropositionalEquality  using ( refl )

-- Imports from the Agda Universal Algebra Library ----------------------------
open import Classical.Signatures.Monoid            using ( Sig-Monoid ; ∙-Op ; ε-Op )
open import Classical.Equations                    using ( Associative ; LeftIdentity ; RightIdentity )
open import Overture.Terms {𝑆 = Sig-Monoid}        using ( Term )

data Eq-Monoid : Type where
  assoc idˡ idʳ : Eq-Monoid

Th-Monoid : Eq-Monoid → Term (Fin 3) × Term (Fin 3)
Th-Monoid assoc  = Associative    ∙-Op       refl 0F 1F 2F
Th-Monoid idˡ    = LeftIdentity   ∙-Op ε-Op  refl refl 0F
Th-Monoid idʳ    = RightIdentity  ∙-Op ε-Op  refl refl 0F