mathlib documentation

order.compactly_generated

Compactness properties for complete lattices #

For complete lattices, there are numerous equivalent ways to express the fact that the relation > is well-founded. In this file we define three especially-useful characterisations and provide proofs that they are indeed equivalent to well-foundedness.

Main definitions #

complete_lattice.is_sup_closed_compact
complete_lattice.is_Sup_finite_compact
complete_lattice.is_compact_element
complete_lattice.is_compactly_generated

Main results #

The main result is that the following four conditions are equivalent for a complete lattice:

well_founded (>)
complete_lattice.is_sup_closed_compact
complete_lattice.is_Sup_finite_compact
∀ k, complete_lattice.is_compact_element k

This is demonstrated by means of the following four lemmas:

We also show well-founded lattices are compactly generated (complete_lattice.compactly_generated_of_well_founded).

References #

G. Călugăreanu, Lattice Concepts of Module Theory

Tags #

complete lattice, well-founded, compact

def complete_lattice.is_sup_closed_compact (α : Type u_1) [complete_lattice α] :

Prop

A compactness property for a complete lattice is that any sup-closed non-empty subset contains its Sup.

Equations

complete_lattice.is_sup_closed_compact α = ∀ (s : set α), s.nonempty → (∀ (a : α), a ∈ s → ∀ (b : α), b ∈ s → a ⊔ b ∈ s) → Sup s ∈ s

def complete_lattice.is_Sup_finite_compact (α : Type u_1) [complete_lattice α] :

Prop

A compactness property for a complete lattice is that any subset has a finite subset with the same Sup.

Equations

complete_lattice.is_Sup_finite_compact α = ∀ (s : set α), ∃ (t : finset α), ↑t ⊆ s ∧ Sup s = t.sup id

def complete_lattice.is_compact_element {α : Type u_1} [complete_lattice α] (k : α) :

Prop

An element k of a complete lattice is said to be compact if any set with Sup above k has a finite subset with Sup above k. Such an element is also called "finite" or "S-compact".

Equations

complete_lattice.is_compact_element k = ∀ (s : set α), k ≤ Sup s → (∃ (t : finset α), ↑t ⊆ s ∧ k ≤ t.sup id)

theorem complete_lattice.is_compact_element_iff_le_of_directed_Sup_le (α : Type u_1) [complete_lattice α] (k : α) :

complete_lattice.is_compact_element k ↔ ∀ (s : set α), s.nonempty → directed_on has_le.le s → k ≤ Sup s → (∃ (x : α), x ∈ s ∧ k ≤ x)

An element k is compact if and only if any directed set with Sup above k already got above k at some point in the set.

theorem complete_lattice.is_compact_element.directed_Sup_lt_of_lt {α : Type u_1} [complete_lattice α] {k : α} (hk : complete_lattice.is_compact_element k) {s : set α} (hemp : s.nonempty) (hdir : directed_on has_le.le s) (hbelow : ∀ (x : α), x ∈ s → x < k) :

Sup s < k

A compact element k has the property that any directed set lying strictly below k has its Sup strictly below k.

theorem complete_lattice.finset_sup_compact_of_compact {α : Type u_1} {β : Type u_2} [complete_lattice α] {f : β → α} (s : finset β) (h : ∀ (x : β), x ∈ s → complete_lattice.is_compact_element (f x)) :

complete_lattice.is_compact_element (s.sup f)

theorem complete_lattice.well_founded.is_Sup_finite_compact (α : Type u_1) [complete_lattice α] (h : well_founded gt) :

complete_lattice.is_Sup_finite_compact α

theorem complete_lattice.is_Sup_finite_compact.is_sup_closed_compact (α : Type u_1) [complete_lattice α] (h : complete_lattice.is_Sup_finite_compact α) :

complete_lattice.is_sup_closed_compact α

theorem complete_lattice.is_sup_closed_compact.well_founded (α : Type u_1) [complete_lattice α] (h : complete_lattice.is_sup_closed_compact α) :

well_founded gt

theorem complete_lattice.is_Sup_finite_compact_iff_all_elements_compact (α : Type u_1) [complete_lattice α] :

complete_lattice.is_Sup_finite_compact α ↔ ∀ (k : α), complete_lattice.is_compact_element k

theorem complete_lattice.well_founded_characterisations (α : Type u_1) [complete_lattice α] :

[well_founded gt, complete_lattice.is_Sup_finite_compact α, complete_lattice.is_sup_closed_compact α, ∀ (k : α), complete_lattice.is_compact_element k].tfae

theorem complete_lattice.well_founded_iff_is_Sup_finite_compact (α : Type u_1) [complete_lattice α] :

well_founded gt ↔ complete_lattice.is_Sup_finite_compact α

theorem complete_lattice.is_Sup_finite_compact_iff_is_sup_closed_compact (α : Type u_1) [complete_lattice α] :

complete_lattice.is_Sup_finite_compact α ↔ complete_lattice.is_sup_closed_compact α

theorem complete_lattice.is_sup_closed_compact_iff_well_founded (α : Type u_1) [complete_lattice α] :

complete_lattice.is_sup_closed_compact α ↔ well_founded gt

theorem is_Sup_finite_compact.well_founded (α : Type u_1) [complete_lattice α] :

complete_lattice.is_Sup_finite_compact α → well_founded gt

Alias of well_founded_iff_is_Sup_finite_compact.

theorem is_sup_closed_compact.is_Sup_finite_compact (α : Type u_1) [complete_lattice α] :

complete_lattice.is_sup_closed_compact α → complete_lattice.is_Sup_finite_compact α

Alias of is_Sup_finite_compact_iff_is_sup_closed_compact.

theorem well_founded.is_sup_closed_compact (α : Type u_1) [complete_lattice α] :

well_founded gt → complete_lattice.is_sup_closed_compact α

Alias of is_sup_closed_compact_iff_well_founded.

@[class]

structure is_compactly_generated (α : Type u_2) [complete_lattice α] :

Prop

exists_Sup_eq : ∀ (x : α), ∃ (s : set α), (∀ (x : α), x ∈ s → complete_lattice.is_compact_element x) ∧ Sup s = x

A complete lattice is said to be compactly generated if any element is the Sup of compact elements.

Instances

submodule.is_compactly_generated

@[simp]

theorem Sup_compact_le_eq {α : Type u_1} [complete_lattice α] [is_compactly_generated α] (b : α) :

Sup {c : α | complete_lattice.is_compact_element c ∧ c ≤ b} = b

@[simp]

theorem Sup_compact_eq_top {α : Type u_1} [complete_lattice α] [is_compactly_generated α] :

Sup {a : α | complete_lattice.is_compact_element a} = ⊤

theorem le_iff_compact_le_imp {α : Type u_1} [complete_lattice α] [is_compactly_generated α] {a b : α} :

a ≤ b ↔ ∀ (c : α), complete_lattice.is_compact_element c → c ≤ a → c ≤ b

theorem inf_Sup_eq_of_directed_on {α : Type u_1} [complete_lattice α] [is_compactly_generated α] {a : α} {s : set α} (h : directed_on has_le.le s) :

a ⊓ Sup s = ⨆ (b : α) (H : b ∈ s), a ⊓ b

This property is sometimes referred to as α being upper continuous.

theorem inf_Sup_eq_supr_inf_sup_finset {α : Type u_1} [complete_lattice α] [is_compactly_generated α] {a : α} {s : set α} :

a ⊓ Sup s = ⨆ (t : finset α) (H : ↑t ⊆ s), a ⊓ t.sup id

This property is equivalent to α being upper continuous.

theorem complete_lattice.set_independent_iff_finite {α : Type u_1} [complete_lattice α] [is_compactly_generated α] {s : set α} :

complete_lattice.set_independent s ↔ ∀ (t : finset α), ↑t ⊆ s → complete_lattice.set_independent ↑t

theorem complete_lattice.set_independent_Union_of_directed {α : Type u_1} [complete_lattice α] [is_compactly_generated α] {η : Type u_2} {s : η → set α} (hs : directed has_subset.subset s) (h : ∀ (i : η), complete_lattice.set_independent (s i)) :

complete_lattice.set_independent (⋃ (i : η), s i)

theorem complete_lattice.independent_sUnion_of_directed {α : Type u_1} [complete_lattice α] [is_compactly_generated α] {s : set (set α)} (hs : directed_on has_subset.subset s) (h : ∀ (a : set α), a ∈ s → complete_lattice.set_independent a) :

complete_lattice.set_independent (⋃₀s)

theorem complete_lattice.compactly_generated_of_well_founded {α : Type u_1} [complete_lattice α] (h : well_founded gt) :

is_compactly_generated α

theorem complete_lattice.Iic_coatomic_of_compact_element {α : Type u_1} [complete_lattice α] {k : α} (h : complete_lattice.is_compact_element k) :

is_coatomic ↥(set.Iic k)

A compact element k has the property that any b < k lies below a "maximal element below k", which is to say [⊥, k] is coatomic.

theorem complete_lattice.coatomic_of_top_compact {α : Type u_1} [complete_lattice α] (h : complete_lattice.is_compact_element ⊤) :

@[protected, instance]

def is_atomic_of_is_complemented {α : Type u_1} [complete_lattice α] [is_modular_lattice α] [is_compactly_generated α] [is_complemented α] :

@[protected, instance]

def is_atomistic_of_is_complemented {α : Type u_1} [complete_lattice α] [is_modular_lattice α] [is_compactly_generated α] [is_complemented α] :

is_atomistic α

See Lemma 5.1, Călugăreanu

theorem is_complemented_of_Sup_atoms_eq_top {α : Type u_1} [complete_lattice α] [is_modular_lattice α] [is_compactly_generated α] (h : Sup {a : α | is_atom a} = ⊤) :

is_complemented α

See Theorem 6.6, Călugăreanu

theorem is_complemented_of_is_atomistic {α : Type u_1} [complete_lattice α] [is_modular_lattice α] [is_compactly_generated α] [is_atomistic α] :

is_complemented α

See Theorem 6.6, Călugăreanu

theorem is_complemented_iff_is_atomistic {α : Type u_1} [complete_lattice α] [is_modular_lattice α] [is_compactly_generated α] :

is_complemented α ↔ is_atomistic α