Homomorphisms and group actions

@[reducible, inline]

abbrev Function.Surjective.mulActionLeft {R : Type u_4} {S : Type u_5} {M : Type u_6} [Monoid R] [MulAction R M] [Monoid S] [SMul S M] (f : R →* S) (hf : Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c • x = c • x) : MulAction S M

Push forward the action of R on M along a compatible surjective map f : R →* S.

See also Function.Surjective.distribMulActionLeft and Function.Surjective.moduleLeft.

Equations

• Function.Surjective.mulActionLeft f hf hsmul = { smul := fun (x1 : S) (x2 : M) => x1 • x2, one_smul := ⋯, mul_smul := ⋯ }

@[reducible, inline]

abbrev Function.Surjective.addActionLeft {R : Type u_4} {S : Type u_5} {M : Type u_6} [AddMonoid R] [AddAction R M] [AddMonoid S] [VAdd S M] (f : R →+ S) (hf : Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c +ᵥ x = c +ᵥ x) : AddAction S M

Push forward the action of R on M along a compatible surjective map f : R →+ S.

Equations

• Function.Surjective.addActionLeft f hf hsmul = { vadd := fun (x1 : S) (x2 : M) => x1 +ᵥ x2, zero_vadd := ⋯, add_vadd := ⋯ }

@[reducible, inline]

abbrev MulAction.compHom {M : Type u_1} {N : Type u_2} (α : Type u_3) [Monoid M] [MulAction M α] [Monoid N] (g : N →* M) : MulAction N α

A multiplicative action of M on α and a monoid homomorphism N → M induce a multiplicative action of N on α.

See note [reducible non-instances].

Equations

• MulAction.compHom α g = { smul := SMul.comp.smul ⇑g, one_smul := ⋯, mul_smul := ⋯ }

@[reducible, inline]

abbrev AddAction.compHom {M : Type u_1} {N : Type u_2} (α : Type u_3) [AddMonoid M] [AddAction M α] [AddMonoid N] (g : N →+ M) : AddAction N α

An additive action of M on α and an additive monoid homomorphism N → M induce an additive action of N on α.

See note [reducible non-instances].

Equations

• AddAction.compHom α g = { vadd := VAdd.comp.vadd ⇑g, zero_vadd := ⋯, add_vadd := ⋯ }

theorem MulAction.compHom_smul_def {E : Type u_4} {F : Type u_5} {G : Type u_6} [Monoid E] [Monoid F] [MulAction F G] (f : E →* F) (a : E) (x : G) : a • x = f a • x

theorem AddAction.compHom_vadd_def {E : Type u_4} {F : Type u_5} {G : Type u_6} [AddMonoid E] [AddMonoid F] [AddAction F G] (f : E →+ F) (a : E) (x : G) : a +ᵥ x = f a +ᵥ x

theorem MulAction.isPretransitive_compHom {E : Type u_4} {F : Type u_5} {G : Type u_6} [Monoid E] [Monoid F] [MulAction F G] [IsPretransitive F G] {f : E →* F} (hf : Function.Surjective ⇑f) : IsPretransitive E G

If an action is transitive, then composing this action with a surjective homomorphism gives again a transitive action.

theorem AddAction.isPretransitive_compHom {E : Type u_4} {F : Type u_5} {G : Type u_6} [AddMonoid E] [AddMonoid F] [AddAction F G] [IsPretransitive F G] {f : E →+ F} (hf : Function.Surjective ⇑f) : IsPretransitive E G

theorem MulAction.IsPretransitive.of_compHom {M : Type u_4} {N : Type u_5} {α : Type u_6} [Monoid M] [Monoid N] [MulAction N α] (f : M →* N) [h : IsPretransitive M α] : IsPretransitive N α

theorem AddAction.IsPretransitive.of_compHom {M : Type u_4} {N : Type u_5} {α : Type u_6} [AddMonoid M] [AddMonoid N] [AddAction N α] (f : M →+ N) [h : IsPretransitive M α] : IsPretransitive N α

def MonoidHom.smulOneHom {M : Type u_4} {N : Type u_5} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] : M →* N

If the multiplicative action of M on N is compatible with multiplication on N, then fun x ↦ x • 1 is a monoid homomorphism from M to N.

Equations

• MonoidHom.smulOneHom = { toFun := fun (x : M) => x • 1, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHom.vaddZeroHom {M : Type u_4} {N : Type u_5} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] : M →+ N

If the additive action of M on N is compatible with addition on N, then fun x ↦ x +ᵥ 0 is an additive monoid homomorphism from M to N.

Equations

• AddMonoidHom.vaddZeroHom = { toFun := fun (x : M) => x +ᵥ 0, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem AddMonoidHom.vaddZeroHom_apply {M : Type u_4} {N : Type u_5} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] (x : M) : vaddZeroHom x = x +ᵥ 0

@[simp]

theorem MonoidHom.smulOneHom_apply {M : Type u_4} {N : Type u_5} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] (x : M) : smulOneHom x = x • 1

def monoidHomEquivMulActionIsScalarTower (M : Type u_4) (N : Type u_5) [Monoid M] [Monoid N] : (M →* N) ≃ { _inst : MulAction M N // IsScalarTower M N N }

A monoid homomorphism between two monoids M and N can be equivalently specified by a multiplicative action of M on N that is compatible with the multiplication on N.

Equations

• One or more equations did not get rendered due to their size.

def addMonoidHomEquivAddActionVAddAssocClass (M : Type u_4) (N : Type u_5) [AddMonoid M] [AddMonoid N] : (M →+ N) ≃ { _inst : AddAction M N // VAddAssocClass M N N }

A monoid homomorphism between two additive monoids M and N can be equivalently specified by an additive action of M on N that is compatible with the addition on N.

Equations

• One or more equations did not get rendered due to their size.