Modular Lattices #

This file defines Modular Lattices, a kind of lattice useful in algebra. For examples, look to the subobject lattices of abelian groups, submodules, and ideals, or consider any distributive lattice.

Main Definitions #

is_modular_lattice defines a modular lattice to be one such that x ≤ z → (x ⊔ y) ⊓ z ≤ x ⊔ (y ⊓ z)
inf_Icc_order_iso_Icc_sup gives an order isomorphism between the intervals [a ⊓ b, a] and [b, a ⊔ b]. This corresponds to the diamond (or second) isomorphism theorems of algebra.

Main Results #

is_modular_lattice_iff_inf_sup_inf_assoc: Modularity is equivalent to the inf_sup_inf_assoc: (x ⊓ z) ⊔ (y ⊓ z) = ((x ⊓ z) ⊔ y) ⊓ z
distrib_lattice.is_modular_lattice: Distributive lattices are modular.

To do #

Relate atoms and coatoms in modular lattices

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@[class]

structure is_modular_lattice (α : Type u_2) [lattice α] :

Prop

sup_inf_le_assoc_of_le : ∀ {x : α} (y : α) {z : α}, x ≤ z → (x ⊔ y) ⊓ z ≤ x ⊔ y ⊓ z

A modular lattice is one with a limited associativity between ⊓ and ⊔.

Instances

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theorem sup_inf_assoc_of_le {α : Type u_1} [lattice α] [is_modular_lattice α] {x : α} (y : α) {z : α} (h : x ≤ z) :

(x ⊔ y) ⊓ z = x ⊔ y ⊓ z

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theorem is_modular_lattice.inf_sup_inf_assoc {α : Type u_1} [lattice α] [is_modular_lattice α] {x y z : α} :

x ⊓ z ⊔ y ⊓ z = (x ⊓ z ⊔ y) ⊓ z

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theorem inf_sup_assoc_of_le {α : Type u_1} [lattice α] [is_modular_lattice α] {x : α} (y : α) {z : α} (h : z ≤ x) :

x ⊓ y ⊔ z = x ⊓ (y ⊔ z)

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@[protected, instance]

def order_dual.is_modular_lattice {α : Type u_1} [lattice α] [is_modular_lattice α] :

is_modular_lattice (order_dual α)

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theorem is_modular_lattice.sup_inf_sup_assoc {α : Type u_1} [lattice α] [is_modular_lattice α] {x y z : α} :

(x ⊔ z) ⊓ (y ⊔ z) = (x ⊔ z) ⊓ y ⊔ z

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theorem eq_of_le_of_inf_le_of_sup_le {α : Type u_1} [lattice α] [is_modular_lattice α] {x y z : α} (hxy : x ≤ y) (hinf : y ⊓ z ≤ x ⊓ z) (hsup : y ⊔ z ≤ x ⊔ z) :

x = y

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theorem sup_lt_sup_of_lt_of_inf_le_inf {α : Type u_1} [lattice α] [is_modular_lattice α] {x y z : α} (hxy : x < y) (hinf : y ⊓ z ≤ x ⊓ z) :

x ⊔ z < y ⊔ z

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theorem inf_lt_inf_of_lt_of_sup_le_sup {α : Type u_1} [lattice α] [is_modular_lattice α] {x y z : α} (hxy : x < y) (hinf : y ⊔ z ≤ x ⊔ z) :

x ⊓ z < y ⊓ z

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theorem well_founded_lt_exact_sequence {α : Type u_1} [lattice α] [is_modular_lattice α] {β : Type u_2} {γ : Type u_3} [partial_order β] [partial_order γ] (h₁ : well_founded has_lt.lt) (h₂ : well_founded has_lt.lt) (K : α) (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : galois_coinsertion f₁ f₂) (gi : galois_insertion g₂ g₁) (hf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K) (hg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K) :

well_founded has_lt.lt

A generalization of the theorem that if N is a submodule of M and N and M / N are both Artinian, then M is Artinian.

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theorem well_founded_gt_exact_sequence {α : Type u_1} [lattice α] [is_modular_lattice α] {β : Type u_2} {γ : Type u_3} [partial_order β] [partial_order γ] (h₁ : well_founded gt) (h₂ : well_founded gt) (K : α) (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : galois_coinsertion f₁ f₂) (gi : galois_insertion g₂ g₁) (hf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K) (hg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K) :

well_founded gt

A generalization of the theorem that if N is a submodule of M and N and M / N are both Noetherian, then M is Noetherian.

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def inf_Icc_order_iso_Icc_sup {α : Type u_1} [lattice α] [is_modular_lattice α] (a b : α) :

↥(set.Icc (a ⊓ b) a) ≃o ↥(set.Icc b (a ⊔ b))

The diamond isomorphism between the intervals [a ⊓ b, a] and [b, a ⊔ b]

Equations

inf_Icc_order_iso_Icc_sup a b = {to_equiv := {to_fun := λ (x : ↥(set.Icc (a ⊓ b) a)), ⟨↑x ⊔ b, _⟩, inv_fun := λ (x : ↥(set.Icc b (a ⊔ b))), ⟨a ⊓ ↑x, _⟩, left_inv := _, right_inv := _}, map_rel_iff' := _}

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def is_compl.Iic_order_iso_Ici {α : Type u_1} [lattice α] [bounded_order α] [is_modular_lattice α] {a b : α} (h : is_compl a b) :

↥(set.Iic a) ≃o ↥(set.Ici b)

The diamond isomorphism between the intervals set.Iic a and set.Ici b.

Equations

h.Iic_order_iso_Ici = (order_iso.set_congr (set.Iic a) (set.Icc (a ⊓ b) a) _).trans ((inf_Icc_order_iso_Icc_sup a b).trans (order_iso.set_congr (set.Icc b (a ⊔ b)) (set.Ici b) _))

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theorem is_modular_lattice_iff_inf_sup_inf_assoc {α : Type u_1} [lattice α] :

is_modular_lattice α ↔ ∀ (x y z : α), x ⊓ z ⊔ y ⊓ z = (x ⊓ z ⊔ y) ⊓ z

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@[protected, instance]

def distrib_lattice.is_modular_lattice {α : Type u_1} [distrib_lattice α] :

is_modular_lattice α

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theorem disjoint.disjoint_sup_right_of_disjoint_sup_left {α : Type u_1} [lattice α] [bounded_order α] [is_modular_lattice α] {a b c : α} (h : disjoint a b) (hsup : disjoint (a ⊔ b) c) :

disjoint a (b ⊔ c)

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theorem disjoint.disjoint_sup_left_of_disjoint_sup_right {α : Type u_1} [lattice α] [bounded_order α] [is_modular_lattice α] {a b c : α} (h : disjoint b c) (hsup : disjoint a (b ⊔ c)) :

disjoint (a ⊔ b) c

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@[protected, instance]

def is_modular_lattice.is_modular_lattice_Iic {α : Type u_1} [lattice α] [is_modular_lattice α] {a : α} :

is_modular_lattice ↥(set.Iic a)

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@[protected, instance]

def is_modular_lattice.is_modular_lattice_Ici {α : Type u_1} [lattice α] [is_modular_lattice α] {a : α} :

is_modular_lattice ↥(set.Ici a)

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@[protected, instance]

def is_modular_lattice.is_complemented_Iic {α : Type u_1} [lattice α] [is_modular_lattice α] {a : α} [bounded_order α] [is_complemented α] :

is_complemented ↥(set.Iic a)

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@[protected, instance]

def is_modular_lattice.is_complemented_Ici {α : Type u_1} [lattice α] [is_modular_lattice α] {a : α} [bounded_order α] [is_complemented α] :

is_complemented ↥(set.Ici a)