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[ add ] left- and right- actions induced by a Magma homomorphism #2894

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[ add ] left- and right- actions induced by a Magma homomorphism #2894

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bcef47e
[ add ] left- and right- actions induced by a `Magma` homomorphism
jamesmckinna Nov 24, 2025
299caba
fix: exports (temporary)
jamesmckinna Nov 24, 2025
97c926f
rename: exports
jamesmckinna Nov 24, 2025
c760a5d
rename: hide renamed exports
jamesmckinna Nov 24, 2025
bb18eca
rename: `Algebra.Module.Morphism.Structure` exports
jamesmckinna Nov 25, 2025
3509702
rename: `Algebra.Lattice.Morphism.Structures` exports
jamesmckinna Nov 25, 2025
453749b
add: re-export multiplicative actions
jamesmckinna Nov 25, 2025
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6 changes: 6 additions & 0 deletions CHANGELOG.md
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Expand Up @@ -151,6 +151,12 @@ Additions to existing modules
⊕-∧-booleanRing : BooleanRing _ _
```

* In `Algebra.Morphism.Structures.IsMagmaHomomorphism`:
```agda
⟦_⟧∙_ : A → B → B
_∙⟦_⟧ : B → A → B
```

* In `Algebra.Properties.RingWithoutOne`:
```agda
[-x][-y]≈xy : ∀ x y → - x * - y ≈ x * y
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1 change: 1 addition & 0 deletions src/Algebra/Lattice/Morphism/Structures.agda
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Expand Up @@ -87,6 +87,7 @@ module LatticeMorphisms (L₁ : RawLattice a ℓ₁) (L₂ : RawLattice b ℓ₂

open ∧.IsMagmaMonomorphism ∧-isMagmaMonomorphism public
using (isRelMonomorphism)
renaming (⟦_⟧∙_ to ⟦_⟧∧_; _∙⟦_⟧ to _∧⟦_⟧)


record IsLatticeIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
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10 changes: 5 additions & 5 deletions src/Algebra/Module/Morphism/Structures.agda
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Expand Up @@ -35,7 +35,7 @@ module LeftSemimoduleMorphisms

open IsMonoidHomomorphism +ᴹ-isMonoidHomomorphism public
using (isRelHomomorphism; ⟦⟧-cong)
renaming (isMagmaHomomorphism to +ᴹ-isMagmaHomomorphism; homo to +ᴹ-homo; ε-homo to 0ᴹ-homo)
renaming (isMagmaHomomorphism to +ᴹ-isMagmaHomomorphism; homo to +ᴹ-homo; ε-homo to 0ᴹ-homo; ⟦_⟧∙_ to ⟦_⟧+ᴹ_; _∙⟦_⟧ to _+ᴹ⟦_⟧)

record IsLeftSemimoduleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
field
Expand Down Expand Up @@ -92,7 +92,7 @@ module LeftModuleMorphisms
using (isRelHomomorphism; ⟦⟧-cong)
renaming ( isMagmaHomomorphism to +ᴹ-isMagmaHomomorphism; isMonoidHomomorphism to +ᴹ-isMonoidHomomorphism
; homo to +ᴹ-homo; ε-homo to 0ᴹ-homo; ⁻¹-homo to -ᴹ-homo
)
; ⟦_⟧∙_ to ⟦_⟧+ᴹ_; _∙⟦_⟧ to _+ᴹ⟦_⟧)

isLeftSemimoduleHomomorphism : IsLeftSemimoduleHomomorphism ⟦_⟧
isLeftSemimoduleHomomorphism = record
Expand Down Expand Up @@ -162,7 +162,7 @@ module RightSemimoduleMorphisms

open IsMonoidHomomorphism +ᴹ-isMonoidHomomorphism public
using (isRelHomomorphism; ⟦⟧-cong)
renaming (isMagmaHomomorphism to +ᴹ-isMagmaHomomorphism; homo to +ᴹ-homo; ε-homo to 0ᴹ-homo)
renaming (isMagmaHomomorphism to +ᴹ-isMagmaHomomorphism; homo to +ᴹ-homo; ε-homo to 0ᴹ-homo; ⟦_⟧∙_ to ⟦_⟧+ᴹ_; _∙⟦_⟧ to _+ᴹ⟦_⟧)

record IsRightSemimoduleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
field
Expand Down Expand Up @@ -219,7 +219,7 @@ module RightModuleMorphisms
using (isRelHomomorphism; ⟦⟧-cong)
renaming ( isMagmaHomomorphism to +ᴹ-isMagmaHomomorphism; isMonoidHomomorphism to +ᴹ-isMonoidHomomorphism
; homo to +ᴹ-homo; ε-homo to 0ᴹ-homo; ⁻¹-homo to -ᴹ-homo
)
; ⟦_⟧∙_ to ⟦_⟧+ᴹ_; _∙⟦_⟧ to _+ᴹ⟦_⟧)
isRightSemimoduleHomomorphism : IsRightSemimoduleHomomorphism ⟦_⟧
isRightSemimoduleHomomorphism = record
{ +ᴹ-isMonoidHomomorphism = +ᴹ-isMonoidHomomorphism
Expand Down Expand Up @@ -376,7 +376,7 @@ module BimoduleMorphisms
using (isRelHomomorphism; ⟦⟧-cong)
renaming ( isMagmaHomomorphism to +ᴹ-isMagmaHomomorphism; isMonoidHomomorphism to +ᴹ-isMonoidHomomorphism
; homo to +ᴹ-homo; ε-homo to 0ᴹ-homo; ⁻¹-homo to -ᴹ-homo
)
; ⟦_⟧∙_ to ⟦_⟧+ᴹ_; _∙⟦_⟧ to _+ᴹ⟦_⟧)

isBisemimoduleHomomorphism : IsBisemimoduleHomomorphism ⟦_⟧
isBisemimoduleHomomorphism = record
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3 changes: 2 additions & 1 deletion src/Algebra/Morphism/Bundles.agda
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Expand Up @@ -163,7 +163,8 @@ record KleeneAlgebraHomomorphism
semiringHomomorphism = record { isSemiringHomomorphism = isSemiringHomomorphism }

open SemiringHomomorphism semiringHomomorphism public
hiding (*-isMagmaHomomorphism; *-isMonoidHomomorphism)
hiding (*-isMagmaHomomorphism; *-isMonoidHomomorphism
; ⟦_⟧+_; _+⟦_⟧; ⟦_⟧*_; _*⟦_⟧)

------------------------------------------------------------------------
-- Morphisms between RingWithoutOnes
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25 changes: 23 additions & 2 deletions src/Algebra/Morphism/Structures.agda
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Expand Up @@ -89,6 +89,12 @@ module MagmaMorphisms (M₁ : RawMagma a ℓ₁) (M₂ : RawMagma b ℓ₂) wher
open IsRelHomomorphism isRelHomomorphism public
renaming (cong to ⟦⟧-cong)

⟦_⟧∙_ : A → B → B
⟦ x ⟧∙ y = ⟦ x ⟧ ◦ y

_∙⟦_⟧ : B → A → B
y ∙⟦ x ⟧ = y ◦ ⟦ x ⟧


record IsMagmaMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
Expand Down Expand Up @@ -261,14 +267,21 @@ module NearSemiringMorphisms (R₁ : RawNearSemiring a ℓ₁) (R₂ : RawNearSe
*-homo : Homomorphic₂ ⟦_⟧ _*₁_ _*₂_

open +.IsMonoidHomomorphism +-isMonoidHomomorphism public
renaming (homo to +-homo; ε-homo to 0#-homo; isMagmaHomomorphism to +-isMagmaHomomorphism)
renaming (homo to +-homo; ε-homo to 0#-homo
; isMagmaHomomorphism to +-isMagmaHomomorphism
; ⟦_⟧∙_ to ⟦_⟧+_; _∙⟦_⟧ to _+⟦_⟧
)

*-isMagmaHomomorphism : *.IsMagmaHomomorphism ⟦_⟧
*-isMagmaHomomorphism = record
{ isRelHomomorphism = isRelHomomorphism
; homo = *-homo
}

open *.IsMagmaHomomorphism *-isMagmaHomomorphism public
using ()
renaming (⟦_⟧∙_ to ⟦_⟧*_; _∙⟦_⟧ to _*⟦_⟧)

record IsNearSemiringMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isNearSemiringHomomorphism : IsNearSemiringHomomorphism ⟦_⟧
Expand Down Expand Up @@ -429,7 +442,11 @@ module RingWithoutOneMorphisms (R₁ : RawRingWithoutOne a ℓ₁) (R₂ : RawRi
*-homo : Homomorphic₂ ⟦_⟧ _*₁_ _*₂_

open +.IsGroupHomomorphism +-isGroupHomomorphism public
renaming (homo to +-homo; ε-homo to 0#-homo; isMagmaHomomorphism to +-isMagmaHomomorphism; isMonoidHomomorphism to +-isMonoidHomomorphism)
renaming (homo to +-homo; ε-homo to 0#-homo
; isMagmaHomomorphism to +-isMagmaHomomorphism
; isMonoidHomomorphism to +-isMonoidHomomorphism
; ⟦_⟧∙_ to ⟦_⟧+_; _∙⟦_⟧ to _+⟦_⟧
)

isNearSemiringHomomorphism : +*.IsNearSemiringHomomorphism ⟦_⟧
isNearSemiringHomomorphism = record
Expand All @@ -443,6 +460,10 @@ module RingWithoutOneMorphisms (R₁ : RawRingWithoutOne a ℓ₁) (R₂ : RawRi
; homo = *-homo
}

open *.IsMagmaHomomorphism *-isMagmaHomomorphism public
using ()
renaming (⟦_⟧∙_ to ⟦_⟧*_; _∙⟦_⟧ to _*⟦_⟧)

record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isRingWithoutOneHomomorphism : IsRingWithoutOneHomomorphism ⟦_⟧
Expand Down