Unbundled monoid and group homomorphisms

This file is deprecated, and is no longer imported by anything in mathlib other than other deprecated files, and test files. You should not need to import it.

This file defines predicates for unbundled monoid and group homomorphisms. Instead of using this file, please use MonoidHom, defined in Algebra.Hom.Group, with notation →*, for morphisms between monoids or groups. For example use φ : G →* H to represent a group homomorphism between multiplicative groups, and ψ : A →+ B to represent a group homomorphism between additive groups.

Main Definitions

IsMonoidHom (deprecated), IsGroupHom (deprecated)

Tags

IsGroupHom, IsMonoidHom

structure IsAddHom {α : Type u_1} {β : Type u_2} [Add α] [Add β] (f : α → β) : Prop

Predicate for maps which preserve an addition.

• map_add (x y : α) : f (x + y) = f x + f y

  The proposition that f preserves addition.

structure IsMulHom {α : Type u_1} {β : Type u_2} [Mul α] [Mul β] (f : α → β) : Prop

Predicate for maps which preserve a multiplication.

• map_mul (x y : α) : f (x * y) = f x * f y

  The proposition that f preserves multiplication.

theorem IsMulHom.id {α : Type u} [Mul α] : IsMulHom id

The identity map preserves multiplication.

theorem IsAddHom.id {α : Type u} [Add α] : IsAddHom id

The identity map preserves addition

theorem IsMulHom.comp {α : Type u} {β : Type v} [Mul α] [Mul β] {γ : Type u_1} [Mul γ] {f : α → β} {g : β → γ} (hf : IsMulHom f) (hg : IsMulHom g) : IsMulHom (g ∘ f)

The composition of maps which preserve multiplication, also preserves multiplication.

theorem IsAddHom.comp {α : Type u} {β : Type v} [Add α] [Add β] {γ : Type u_1} [Add γ] {f : α → β} {g : β → γ} (hf : IsAddHom f) (hg : IsAddHom g) : IsAddHom (g ∘ f)

The composition of addition preserving maps also preserves addition

theorem IsMulHom.mul {α : Type u_2} {β : Type u_3} [Semigroup α] [CommSemigroup β] {f g : α → β} (hf : IsMulHom f) (hg : IsMulHom g) : IsMulHom fun (a : α) => f a * g a

A product of maps which preserve multiplication, preserves multiplication when the target is commutative.

theorem IsAddHom.add {α : Type u_2} {β : Type u_3} [AddSemigroup α] [AddCommSemigroup β] {f g : α → β} (hf : IsAddHom f) (hg : IsAddHom g) : IsAddHom fun (a : α) => f a + g a

A sum of maps which preserves addition, preserves addition when the target is commutative.

theorem IsMulHom.inv {α : Type u_2} {β : Type u_3} [Mul α] [CommGroup β] {f : α → β} (hf : IsMulHom f) : IsMulHom fun (a : α) => (f a)⁻¹

The inverse of a map which preserves multiplication, preserves multiplication when the target is commutative.

theorem IsAddHom.neg {α : Type u_2} {β : Type u_3} [Add α] [AddCommGroup β] {f : α → β} (hf : IsAddHom f) : IsAddHom fun (a : α) => -f a

The negation of a map which preserves addition, preserves addition when the target is commutative.

structure IsAddMonoidHom {α : Type u} {β : Type v} [AddZeroClass α] [AddZeroClass β] (f : α → β) extends IsAddHom f : Prop

Predicate for additive monoid homomorphisms (deprecated -- use the bundled MonoidHom version).

• map_add (x y : α) : f (x + y) = f x + f y
• map_zero : f 0 = 0

  The proposition that f preserves the additive identity.

structure IsMonoidHom {α : Type u} {β : Type v} [MulOneClass α] [MulOneClass β] (f : α → β) extends IsMulHom f : Prop

Predicate for monoid homomorphisms (deprecated -- use the bundled MonoidHom version).

• map_mul (x y : α) : f (x * y) = f x * f y
• map_one : f 1 = 1

  The proposition that f preserves the multiplicative identity.

def MonoidHom.of {M : Type u_1} {N : Type u_2} {mM : MulOneClass M} {mN : MulOneClass N} {f : M → N} (h : IsMonoidHom f) : M →* N

Interpret a map f : M → N as a homomorphism M →* N.

Equations

• MonoidHom.of h = { toFun := f, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHom.of {M : Type u_1} {N : Type u_2} {mM : AddZeroClass M} {mN : AddZeroClass N} {f : M → N} (h : IsAddMonoidHom f) : M →+ N

Interpret a map f : M → N as a homomorphism M →+ N.

Equations

• AddMonoidHom.of h = { toFun := f, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem MonoidHom.coe_of {M : Type u_1} {N : Type u_2} {mM : MulOneClass M} {mN : MulOneClass N} {f : M → N} (hf : IsMonoidHom f) : ⇑(MonoidHom.of hf) = f

@[simp]

theorem AddMonoidHom.coe_of {M : Type u_1} {N : Type u_2} {mM : AddZeroClass M} {mN : AddZeroClass N} {f : M → N} (hf : IsAddMonoidHom f) : ⇑(AddMonoidHom.of hf) = f

theorem MonoidHom.isMonoidHom_coe {M : Type u_1} {N : Type u_2} {mM : MulOneClass M} {mN : MulOneClass N} (f : M →* N) : IsMonoidHom ⇑f

theorem AddMonoidHom.isAddMonoidHom_coe {M : Type u_1} {N : Type u_2} {mM : AddZeroClass M} {mN : AddZeroClass N} (f : M →+ N) : IsAddMonoidHom ⇑f

theorem MulEquiv.isMulHom {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] (h : M ≃* N) : IsMulHom ⇑h

A multiplicative isomorphism preserves multiplication (deprecated).

theorem AddEquiv.isAddHom {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (h : M ≃+ N) : IsAddHom ⇑h

An additive isomorphism preserves addition (deprecated).

theorem MulEquiv.isMonoidHom {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] (h : M ≃* N) : IsMonoidHom ⇑h

A multiplicative bijection between two monoids is a monoid hom (deprecated -- use MulEquiv.toMonoidHom).

theorem AddEquiv.isAddMonoidHom {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (h : M ≃+ N) : IsAddMonoidHom ⇑h

An additive bijection between two additive monoids is an additive monoid hom (deprecated).

theorem IsMonoidHom.map_mul' {α : Type u} {β : Type v} [MulOneClass α] [MulOneClass β] {f : α → β} (hf : IsMonoidHom f) (x y : α) : f (x * y) = f x * f y

A monoid homomorphism preserves multiplication.

theorem IsAddMonoidHom.map_add' {α : Type u} {β : Type v} [AddZeroClass α] [AddZeroClass β] {f : α → β} (hf : IsAddMonoidHom f) (x y : α) : f (x + y) = f x + f y

An additive monoid homomorphism preserves addition.

theorem IsMonoidHom.inv {α : Type u_1} {β : Type u_2} [MulOneClass α] [CommGroup β] {f : α → β} (hf : IsMonoidHom f) : IsMonoidHom fun (a : α) => (f a)⁻¹

The inverse of a map which preserves multiplication, preserves multiplication when the target is commutative.

theorem IsAddMonoidHom.neg {α : Type u_1} {β : Type u_2} [AddZeroClass α] [AddCommGroup β] {f : α → β} (hf : IsAddMonoidHom f) : IsAddMonoidHom fun (a : α) => -f a

The negation of a map which preserves addition, preserves addition when the target is commutative.

theorem IsMulHom.to_isMonoidHom {α : Type u} {β : Type v} [MulOneClass α] [Group β] {f : α → β} (hf : IsMulHom f) : IsMonoidHom f

A map to a group preserving multiplication is a monoid homomorphism.

theorem IsAddHom.to_isAddMonoidHom {α : Type u} {β : Type v} [AddZeroClass α] [AddGroup β] {f : α → β} (hf : IsAddHom f) : IsAddMonoidHom f

A map to an additive group preserving addition is an additive monoid homomorphism.

theorem IsMonoidHom.id {α : Type u} [MulOneClass α] : IsMonoidHom id

The identity map is a monoid homomorphism.

theorem IsAddMonoidHom.id {α : Type u} [AddZeroClass α] : IsAddMonoidHom id

The identity map is an additive monoid homomorphism.

theorem IsMonoidHom.comp {α : Type u} {β : Type v} [MulOneClass α] [MulOneClass β] {f : α → β} (hf : IsMonoidHom f) {γ : Type u_1} [MulOneClass γ] {g : β → γ} (hg : IsMonoidHom g) : IsMonoidHom (g ∘ f)

The composite of two monoid homomorphisms is a monoid homomorphism.

theorem IsAddMonoidHom.comp {α : Type u} {β : Type v} [AddZeroClass α] [AddZeroClass β] {f : α → β} (hf : IsAddMonoidHom f) {γ : Type u_1} [AddZeroClass γ] {g : β → γ} (hg : IsAddMonoidHom g) : IsAddMonoidHom (g ∘ f)

The composite of two additive monoid homomorphisms is an additive monoid homomorphism.

theorem IsAddMonoidHom.isAddMonoidHom_mul_left {γ : Type u_1} [NonUnitalNonAssocSemiring γ] (x : γ) : IsAddMonoidHom fun (y : γ) => x * y

Left multiplication in a ring is an additive monoid morphism.

theorem IsAddMonoidHom.isAddMonoidHom_mul_right {γ : Type u_1} [NonUnitalNonAssocSemiring γ] (x : γ) : IsAddMonoidHom fun (y : γ) => y * x

Right multiplication in a ring is an additive monoid morphism.

structure IsAddGroupHom {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] (f : α → β) extends IsAddHom f : Prop

Predicate for additive group homomorphism (deprecated -- use bundled MonoidHom).

• map_add (x y : α) : f (x + y) = f x + f y

structure IsGroupHom {α : Type u} {β : Type v} [Group α] [Group β] (f : α → β) extends IsMulHom f : Prop

Predicate for group homomorphisms (deprecated -- use bundled MonoidHom).

• map_mul (x y : α) : f (x * y) = f x * f y

theorem MonoidHom.isGroupHom {G : Type u_1} {H : Type u_2} {x✝ : Group G} {x✝¹ : Group H} (f : G →* H) : IsGroupHom ⇑f

theorem AddMonoidHom.isAddGroupHom {G : Type u_1} {H : Type u_2} {x✝ : AddGroup G} {x✝¹ : AddGroup H} (f : G →+ H) : IsAddGroupHom ⇑f

theorem MulEquiv.isGroupHom {G : Type u_1} {H : Type u_2} {x✝ : Group G} {x✝¹ : Group H} (h : G ≃* H) : IsGroupHom ⇑h

theorem AddEquiv.isAddGroupHom {G : Type u_1} {H : Type u_2} {x✝ : AddGroup G} {x✝¹ : AddGroup H} (h : G ≃+ H) : IsAddGroupHom ⇑h

theorem IsGroupHom.mk' {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : ∀ (x y : α), f (x * y) = f x * f y) : IsGroupHom f

Construct IsGroupHom from its only hypothesis.

theorem IsAddGroupHom.mk' {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : ∀ (x y : α), f (x + y) = f x + f y) : IsAddGroupHom f

Construct IsAddGroupHom from its only hypothesis.

theorem IsGroupHom.id {α : Type u} [Group α] : IsGroupHom id

The identity is a group homomorphism.

theorem IsAddGroupHom.id {α : Type u} [AddGroup α] : IsAddGroupHom id

The identity is an additive group homomorphism.

theorem IsGroupHom.map_mul' {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) (x y : α) : f (x * y) = f x * f y

theorem IsGroupHom.to_isMonoidHom {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) : IsMonoidHom f

A group homomorphism is a monoid homomorphism.

theorem IsAddGroupHom.to_isAddMonoidHom {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) : IsAddMonoidHom f

An additive group homomorphism is an additive monoid homomorphism.

theorem IsGroupHom.map_one {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) : f 1 = 1

A group homomorphism sends 1 to 1.

theorem IsAddGroupHom.map_zero {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) : f 0 = 0

An additive group homomorphism sends 0 to 0.

theorem IsGroupHom.map_inv {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) (a : α) : f a⁻¹ = (f a)⁻¹

A group homomorphism sends inverses to inverses.

theorem IsAddGroupHom.map_neg {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) (a : α) : f (-a) = -f a

An additive group homomorphism sends negations to negations.

theorem IsGroupHom.map_div {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) (a b : α) : f (a / b) = f a / f b

theorem IsAddGroupHom.map_sub {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) (a b : α) : f (a - b) = f a - f b

theorem IsGroupHom.comp {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) {γ : Type u_1} [Group γ] {g : β → γ} (hg : IsGroupHom g) : IsGroupHom (g ∘ f)

The composition of two group homomorphisms is a group homomorphism.

theorem IsAddGroupHom.comp {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) {γ : Type u_1} [AddGroup γ] {g : β → γ} (hg : IsAddGroupHom g) : IsAddGroupHom (g ∘ f)

The composition of two additive group homomorphisms is an additive group homomorphism.

theorem IsGroupHom.injective_iff {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) : Function.Injective f ↔ ∀ (a : α), f a = 1 → a = 1

A group homomorphism is injective iff its kernel is trivial.

theorem IsAddGroupHom.injective_iff {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) : Function.Injective f ↔ ∀ (a : α), f a = 0 → a = 0

An additive group homomorphism is injective if its kernel is trivial.

theorem IsGroupHom.mul {α : Type u_1} {β : Type u_2} [Group α] [CommGroup β] {f g : α → β} (hf : IsGroupHom f) (hg : IsGroupHom g) : IsGroupHom fun (a : α) => f a * g a

The product of group homomorphisms is a group homomorphism if the target is commutative.

theorem IsAddGroupHom.add {α : Type u_1} {β : Type u_2} [AddGroup α] [AddCommGroup β] {f g : α → β} (hf : IsAddGroupHom f) (hg : IsAddGroupHom g) : IsAddGroupHom fun (a : α) => f a + g a

The sum of two additive group homomorphisms is an additive group homomorphism if the target is commutative.

theorem IsGroupHom.inv {α : Type u_1} {β : Type u_2} [Group α] [CommGroup β] {f : α → β} (hf : IsGroupHom f) : IsGroupHom fun (a : α) => (f a)⁻¹

The inverse of a group homomorphism is a group homomorphism if the target is commutative.

theorem IsAddGroupHom.neg {α : Type u_1} {β : Type u_2} [AddGroup α] [AddCommGroup β] {f : α → β} (hf : IsAddGroupHom f) : IsAddGroupHom fun (a : α) => -f a

The negation of an additive group homomorphism is an additive group homomorphism if the target is commutative.

These instances look redundant, because Deprecated.Ring provides IsRingHom for a →+*. Nevertheless these are harmless, and helpful for stripping out dependencies on Deprecated.Ring.

theorem RingHom.to_isMonoidHom {R : Type u_1} {S : Type u_2} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : IsMonoidHom ⇑f

theorem RingHom.to_isAddMonoidHom {R : Type u_1} {S : Type u_2} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : IsAddMonoidHom ⇑f

theorem RingHom.to_isAddGroupHom {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) : IsAddGroupHom ⇑f

theorem Inv.isGroupHom {α : Type u} [CommGroup α] : IsGroupHom Inv.inv

Inversion is a group homomorphism if the group is commutative.

theorem Neg.isAddGroupHom {α : Type u} [AddCommGroup α] : IsAddGroupHom Neg.neg

Negation is an AddGroup homomorphism if the AddGroup is commutative.

theorem IsAddGroupHom.sub {α : Type u_1} {β : Type u_2} [AddGroup α] [AddCommGroup β] {f g : α → β} (hf : IsAddGroupHom f) (hg : IsAddGroupHom g) : IsAddGroupHom fun (a : α) => f a - g a

The difference of two additive group homomorphisms is an additive group homomorphism if the target is commutative.

@[reducible, inline]

abbrev Units.map' {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {f : M → N} (hf : IsMonoidHom f) : Mˣ →* Nˣ

The group homomorphism on units induced by a multiplicative morphism.

Equations

• Units.map' hf = Units.map (MonoidHom.of hf)

@[simp]

theorem Units.coe_map' {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {f : M → N} (hf : IsMonoidHom f) (x : Mˣ) : ↑((Units.map' hf) x) = f ↑x

theorem Units.coe_isMonoidHom {M : Type u_1} [Monoid M] : IsMonoidHom fun (x : Mˣ) => ↑x

theorem IsUnit.map' {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {f : M → N} (hf : IsMonoidHom f) {x : M} (h : IsUnit x) : IsUnit (f x)

theorem Additive.isAddHom {α : Type u} {β : Type v} [Mul α] [Mul β] {f : α → β} (hf : IsMulHom f) : IsAddHom f

theorem Multiplicative.isMulHom {α : Type u} {β : Type v} [Add α] [Add β] {f : α → β} (hf : IsAddHom f) : IsMulHom f

theorem Additive.isAddMonoidHom {α : Type u} {β : Type v} [MulOneClass α] [MulOneClass β] {f : α → β} (hf : IsMonoidHom f) : IsAddMonoidHom f

theorem Multiplicative.isMonoidHom {α : Type u} {β : Type v} [AddZeroClass α] [AddZeroClass β] {f : α → β} (hf : IsAddMonoidHom f) : IsMonoidHom f

theorem Additive.isAddGroupHom {α : Type u} {β : Type v} [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) : IsAddGroupHom f

theorem Multiplicative.isGroupHom {α : Type u} {β : Type v} [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) : IsGroupHom f