Canonical homomorphism from a finite family of monoids #

This file defines the construction of the canonical homomorphism from a family of monoids.

Given a family of morphisms ϕ i : N i →* M for each i : ι where elements in the images of different morphisms commute, we obtain a canonical morphism MonoidHom.noncommPiCoprod : (Π i, N i) →* M that coincides with ϕ

Main definitions #

MonoidHom.noncommPiCoprod : (Π i, N i) →* M is the main homomorphism
Subgroup.noncommPiCoprod : (Π i, H i) →* G is the specialization to H i : Subgroup G and the subgroup embedding.

Main theorems #

MonoidHom.noncommPiCoprod coincides with ϕ i when restricted to N i
MonoidHom.noncommPiCoprod_mrange: The range of MonoidHom.noncommPiCoprod is ⨆ (i : ι), (ϕ i).mrange
MonoidHom.noncommPiCoprod_range: The range of MonoidHom.noncommPiCoprod is ⨆ (i : ι), (ϕ i).range
Subgroup.noncommPiCoprod_range: The range of Subgroup.noncommPiCoprod is ⨆ (i : ι), H i.
MonoidHom.injective_noncommPiCoprod_of_independent: in the case of groups, pi_hom.hom is injective if the ϕ are injective and the ranges of the ϕ are independent.
MonoidHom.independent_range_of_coprime_order: If the N i have coprime orders, then the ranges of the ϕ are independent.
Subgroup.independent_of_coprime_order: If commuting normal subgroups H i have coprime orders, they are independent.

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theorem AddSubgroup.eq_zero_of_noncommSum_eq_zero_of_independent {G : Type u_1} [AddGroup G] {ι : Type u_2} (s : Finset ι) (f : ι → G) (comm : Set.Pairwise ↑s fun a b => AddCommute (f a) (f b)) (K : ι → AddSubgroup G) (hind : CompleteLattice.Independent K) (hmem : ∀ (x : ι), x ∈ s → f x ∈ K x) (heq1 : Finset.noncommSum s f comm = 0) (i : ι) :

i ∈ s → f i = 0

Finset.noncommSum is “injective” in f if f maps into independent subgroups. This generalizes (one direction of) AddSubgroup.disjoint_iff_add_eq_zero.

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theorem Subgroup.eq_one_of_noncommProd_eq_one_of_independent {G : Type u_1} [Group G] {ι : Type u_2} (s : Finset ι) (f : ι → G) (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) (K : ι → Subgroup G) (hind : CompleteLattice.Independent K) (hmem : ∀ (x : ι), x ∈ s → f x ∈ K x) (heq1 : Finset.noncommProd s f comm = 1) (i : ι) :

i ∈ s → f i = 1

Finset.noncommProd is “injective” in f if f maps into independent subgroups. This generalizes (one direction of) Subgroup.disjoint_iff_mul_eq_one.

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theorem AddMonoidHom.noncommPiCoprod.proof_2 {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) (hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)) :

(Finset.noncommSum Finset.univ (fun i => ↑(ϕ i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i)) fun i x j x h => hcomm i j h (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 j)) = 0

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theorem AddMonoidHom.noncommPiCoprod.proof_3 {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) (hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)) (f : (i : ι) → N i) (g : (i : ι) → N i) :

ZeroHom.toFun { toFun := fun f => Finset.noncommSum Finset.univ (fun i => ↑(ϕ i) (f i)) fun i x j x h => hcomm i j h (f i) (f j), map_zero' := (_ : (Finset.noncommSum Finset.univ (fun i => ↑(ϕ i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i)) fun i x j x h => hcomm i j h (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 j)) = 0) } (f + g) = ZeroHom.toFun { toFun := fun f => Finset.noncommSum Finset.univ (fun i => ↑(ϕ i) (f i)) fun i x j x h => hcomm i j h (f i) (f j), map_zero' := (_ : (Finset.noncommSum Finset.univ (fun i => ↑(ϕ i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i)) fun i x j x h => hcomm i j h (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 j)) = 0) } f + ZeroHom.toFun { toFun := fun f => Finset.noncommSum Finset.univ (fun i => ↑(ϕ i) (f i)) fun i x j x h => hcomm i j h (f i) (f j), map_zero' := (_ : (Finset.noncommSum Finset.univ (fun i => ↑(ϕ i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i)) fun i x j x h => hcomm i j h (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 i) (OfNat.ofNat ((i : ι) → N i) 0 Zero.toOfNat0 j)) = 0) } g

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theorem AddMonoidHom.noncommPiCoprod.proof_1 {M : Type u_2} [AddMonoid M] {ι : Type u_1} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) (hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)) (f : (i : ι) → N i) (i : ι) :

i ∈ ↑Finset.univ → ∀ (j : ι), j ∈ ↑Finset.univ → i ≠ j → AddCommute (↑(ϕ i) (f i)) (↑(ϕ j) (f j))

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def AddMonoidHom.noncommPiCoprod {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) (hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)) :

((i : ι) → N i) →+ M

The canonical homomorphism from a family of additive monoids. See also LinearMap.lsum for a linear version without the commutativity assumption.

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def MonoidHom.noncommPiCoprod {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) (hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute (↑(ϕ i) x) (↑(ϕ j) y)) :

((i : ι) → N i) →* M

The canonical homomorphism from a family of monoids.

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

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@[simp]

theorem AddMonoidHom.noncommPiCoprod_single {M : Type u_1} [AddMonoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)} (i : ι) (y : N i) :

↑(AddMonoidHom.noncommPiCoprod ϕ hcomm) (Pi.single i y) = ↑(ϕ i) y

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@[simp]

theorem MonoidHom.noncommPiCoprod_mulSingle {M : Type u_1} [Monoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute (↑(ϕ i) x) (↑(ϕ j) y)} (i : ι) (y : N i) :

↑(MonoidHom.noncommPiCoprod ϕ hcomm) (Pi.mulSingle i y) = ↑(ϕ i) y

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def AddMonoidHom.noncommPiCoprodEquiv {M : Type u_1} [AddMonoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] :

{ ϕ // Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y) } ≃ (((i : ι) → N i) →+ M)

The universal property of AddMonoidHom.noncommPiCoprod

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theorem AddMonoidHom.noncommPiCoprodEquiv.proof_4 {M : Type u_1} [AddMonoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (f : ((i : ι) → N i) →+ M) :

(fun ϕ => AddMonoidHom.noncommPiCoprod ↑ϕ (_ : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(↑ϕ i) x) (↑(↑ϕ j) y))) ((fun f => { val := fun i => AddMonoidHom.comp f (AddMonoidHom.single N i), property := (_ : ∀ (i j : ι), i ≠ j → ∀ (x : N i) (y : N j), AddCommute (↑f (Pi.single i x)) (↑f (Pi.single j y))) }) f) = f

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theorem AddMonoidHom.noncommPiCoprodEquiv.proof_3 {M : Type u_1} [AddMonoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : { ϕ // Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y) }) :

(fun f => { val := fun i => AddMonoidHom.comp f (AddMonoidHom.single N i), property := (_ : ∀ (i j : ι), i ≠ j → ∀ (x : N i) (y : N j), AddCommute (↑f (Pi.single i x)) (↑f (Pi.single j y))) }) ((fun ϕ => AddMonoidHom.noncommPiCoprod ↑ϕ (_ : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(↑ϕ i) x) (↑(↑ϕ j) y))) ϕ) = ϕ

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theorem AddMonoidHom.noncommPiCoprodEquiv.proof_1 {M : Type u_2} [AddMonoid M] {ι : Type u_1} {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : { ϕ // Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y) }) :

Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(↑ϕ i) x) (↑(↑ϕ j) y)

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theorem AddMonoidHom.noncommPiCoprodEquiv.proof_2 {M : Type u_2} [AddMonoid M] {ι : Type u_1} [hdec : DecidableEq ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (f : ((i : ι) → N i) →+ M) (i : ι) (j : ι) (hij : i ≠ j) (x : N i) (y : N j) :

AddCommute (↑f (Pi.single i x)) (↑f (Pi.single j y))

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def MonoidHom.noncommPiCoprodEquiv {M : Type u_1} [Monoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] :

{ ϕ // Pairwise fun i j => ∀ (x : N i) (y : N j), Commute (↑(ϕ i) x) (↑(ϕ j) y) } ≃ (((i : ι) → N i) →* M)

The universal property of MonoidHom.noncommPiCoprod

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theorem AddMonoidHom.noncommPiCoprod_mrange {M : Type u_1} [AddMonoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)} :

AddMonoidHom.mrange (AddMonoidHom.noncommPiCoprod ϕ hcomm) = ⨆ (i : ι), AddMonoidHom.mrange (ϕ i)

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theorem MonoidHom.noncommPiCoprod_mrange {M : Type u_1} [Monoid M] {ι : Type u_2} [hdec : DecidableEq ι] [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute (↑(ϕ i) x) (↑(ϕ j) y)} :

MonoidHom.mrange (MonoidHom.noncommPiCoprod ϕ hcomm) = ⨆ (i : ι), MonoidHom.mrange (ϕ i)

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theorem AddMonoidHom.noncommPiCoprod_range {G : Type u_1} [AddGroup G] {ι : Type u_2} [hdec : DecidableEq ι] [hfin : Fintype ι] {H : ι → Type u_3} [(i : ι) → AddGroup (H i)] (ϕ : (i : ι) → H i →+ G) {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)} :

AddMonoidHom.range (AddMonoidHom.noncommPiCoprod ϕ hcomm) = ⨆ (i : ι), AddMonoidHom.range (ϕ i)

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theorem MonoidHom.noncommPiCoprod_range {G : Type u_1} [Group G] {ι : Type u_2} [hdec : DecidableEq ι] [hfin : Fintype ι] {H : ι → Type u_3} [(i : ι) → Group (H i)] (ϕ : (i : ι) → H i →* G) {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)} :

MonoidHom.range (MonoidHom.noncommPiCoprod ϕ hcomm) = ⨆ (i : ι), MonoidHom.range (ϕ i)

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theorem AddMonoidHom.injective_noncommPiCoprod_of_independent {G : Type u_1} [AddGroup G] {ι : Type u_2} [hfin : Fintype ι] {H : ι → Type u_3} [(i : ι) → AddGroup (H i)] (ϕ : (i : ι) → H i →+ G) {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)} (hind : CompleteLattice.Independent fun i => AddMonoidHom.range (ϕ i)) (hinj : ∀ (i : ι), Function.Injective ↑(ϕ i)) :

Function.Injective ↑(AddMonoidHom.noncommPiCoprod ϕ hcomm)

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theorem MonoidHom.injective_noncommPiCoprod_of_independent {G : Type u_1} [Group G] {ι : Type u_2} [hfin : Fintype ι] {H : ι → Type u_3} [(i : ι) → Group (H i)] (ϕ : (i : ι) → H i →* G) {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)} (hind : CompleteLattice.Independent fun i => MonoidHom.range (ϕ i)) (hinj : ∀ (i : ι), Function.Injective ↑(ϕ i)) :

Function.Injective ↑(MonoidHom.noncommPiCoprod ϕ hcomm)

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theorem AddMonoidHom.independent_range_of_coprime_order {G : Type u_1} [AddGroup G] {ι : Type u_2} [hdec : DecidableEq ι] {H : ι → Type u_3} [(i : ι) → AddGroup (H i)] (ϕ : (i : ι) → H i →+ G) (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), AddCommute (↑(ϕ i) x) (↑(ϕ j) y)) [Finite ι] [(i : ι) → Fintype (H i)] (hcoprime : ∀ (i j : ι), i ≠ j → Nat.Coprime (Fintype.card (H i)) (Fintype.card (H j))) :

CompleteLattice.Independent fun i => AddMonoidHom.range (ϕ i)

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theorem MonoidHom.independent_range_of_coprime_order {G : Type u_1} [Group G] {ι : Type u_2} [hdec : DecidableEq ι] {H : ι → Type u_3} [(i : ι) → Group (H i)] (ϕ : (i : ι) → H i →* G) (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)) [Finite ι] [(i : ι) → Fintype (H i)] (hcoprime : ∀ (i j : ι), i ≠ j → Nat.Coprime (Fintype.card (H i)) (Fintype.card (H j))) :

CompleteLattice.Independent fun i => MonoidHom.range (ϕ i)

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theorem AddSubgroup.commute_subtype_of_commute {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) (i : ι) (j : ι) (hne : i ≠ j) (x : { x // x ∈ H i }) (y : { x // x ∈ H j }) :

AddCommute (↑(AddSubgroup.subtype (H i)) x) (↑(AddSubgroup.subtype (H j)) y)

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theorem Subgroup.commute_subtype_of_commute {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y) (i : ι) (j : ι) (hne : i ≠ j) (x : { x // x ∈ H i }) (y : { x // x ∈ H j }) :

Commute (↑(Subgroup.subtype (H i)) x) (↑(Subgroup.subtype (H j)) y)

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def AddSubgroup.noncommPiCoprod {G : Type u_1} [AddGroup G] {ι : Type u_2} [hfin : Fintype ι] {H : ι → AddSubgroup G} (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) :

((i : ι) → { x // x ∈ H i }) →+ G

The canonical homomorphism from a family of additive subgroups where elements from different subgroups commute

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theorem AddSubgroup.noncommPiCoprod.proof_1 {G : Type u_2} [AddGroup G] {ι : Type u_1} {H : ι → AddSubgroup G} (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) (i : ι) (j : ι) (hne : i ≠ j) (x : { x // x ∈ H i }) (y : { x // x ∈ H j }) :

AddCommute (↑(AddSubgroup.subtype (H i)) x) (↑(AddSubgroup.subtype (H j)) y)

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def Subgroup.noncommPiCoprod {G : Type u_1} [Group G] {ι : Type u_2} [hfin : Fintype ι] {H : ι → Subgroup G} (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y) :

((i : ι) → { x // x ∈ H i }) →* G

The canonical homomorphism from a family of subgroups where elements from different subgroups commute

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

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@[simp]

theorem AddSubgroup.noncommPiCoprod_single {G : Type u_1} [AddGroup G] {ι : Type u_2} [hdec : DecidableEq ι] [hfin : Fintype ι] {H : ι → AddSubgroup G} {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y} (i : ι) (y : { x // x ∈ H i }) :

↑(AddSubgroup.noncommPiCoprod hcomm) (Pi.single i y) = ↑y

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@[simp]

theorem Subgroup.noncommPiCoprod_mulSingle {G : Type u_1} [Group G] {ι : Type u_2} [hdec : DecidableEq ι] [hfin : Fintype ι] {H : ι → Subgroup G} {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y} (i : ι) (y : { x // x ∈ H i }) :

↑(Subgroup.noncommPiCoprod hcomm) (Pi.mulSingle i y) = ↑y

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theorem AddSubgroup.noncommPiCoprod_range {G : Type u_1} [AddGroup G] {ι : Type u_2} [hdec : DecidableEq ι] [hfin : Fintype ι] {H : ι → AddSubgroup G} {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y} :

AddMonoidHom.range (AddSubgroup.noncommPiCoprod hcomm) = ⨆ (i : ι), H i

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theorem Subgroup.noncommPiCoprod_range {G : Type u_1} [Group G] {ι : Type u_2} [hdec : DecidableEq ι] [hfin : Fintype ι] {H : ι → Subgroup G} {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y} :

MonoidHom.range (Subgroup.noncommPiCoprod hcomm) = ⨆ (i : ι), H i

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theorem AddSubgroup.injective_noncommPiCoprod_of_independent {G : Type u_1} [AddGroup G] {ι : Type u_2} [hfin : Fintype ι] {H : ι → AddSubgroup G} {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y} (hind : CompleteLattice.Independent H) :

Function.Injective ↑(AddSubgroup.noncommPiCoprod hcomm)

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theorem Subgroup.injective_noncommPiCoprod_of_independent {G : Type u_1} [Group G] {ι : Type u_2} [hfin : Fintype ι] {H : ι → Subgroup G} {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y} (hind : CompleteLattice.Independent H) :

Function.Injective ↑(Subgroup.noncommPiCoprod hcomm)

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theorem AddSubgroup.independent_of_coprime_order {G : Type u_1} [AddGroup G] {ι : Type u_2} [hdec : DecidableEq ι] {H : ι → AddSubgroup G} (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) [Finite ι] [(i : ι) → Fintype { x // x ∈ H i }] (hcoprime : ∀ (i j : ι), i ≠ j → Nat.Coprime (Fintype.card { x // x ∈ H i }) (Fintype.card { x // x ∈ H j })) :

CompleteLattice.Independent H

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theorem Subgroup.independent_of_coprime_order {G : Type u_1} [Group G] {ι : Type u_2} [hdec : DecidableEq ι] {H : ι → Subgroup G} (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y) [Finite ι] [(i : ι) → Fintype { x // x ∈ H i }] (hcoprime : ∀ (i j : ι), i ≠ j → Nat.Coprime (Fintype.card { x // x ∈ H i }) (Fintype.card { x // x ∈ H j })) :

CompleteLattice.Independent H