# mathlibdocumentation

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:

• complete_lattice.well_founded.is_Sup_finite_compact
• complete_lattice.is_Sup_finite_compact.is_sup_closed_compact
• complete_lattice.is_sup_closed_compact.well_founded
• complete_lattice.is_Sup_finite_compact_iff_all_elements_compact

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

## Tags #

complete lattice, well-founded, compact

def complete_lattice.is_sup_closed_compact (α : Type u_1)  :
Prop

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

Equations
def complete_lattice.is_Sup_finite_compact (α : Type u_1)  :
Prop

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

Equations
def complete_lattice.is_compact_element {α : Type u_1} (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
theorem complete_lattice.is_compact_element_iff {α : Type u} (k : α) :
∀ (ι : Type u) (s : ι → α), k supr s(∃ (t : finset ι), k t.sup s)
theorem complete_lattice.is_compact_element_iff_le_of_directed_Sup_le (α : Type u_1) (k : α) :
∀ (s : set α), s.nonemptyk (∃ (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.exists_finset_of_le_supr (α : Type u_1) {k : α} {ι : Type u_2} (f : ι → α) (h : k ⨆ (i : ι), f i) :
∃ (s : finset ι), k ⨆ (i : ι) (H : i s), f i
theorem complete_lattice.is_compact_element.directed_Sup_lt_of_lt {α : Type u_1} {k : α} {s : set α} (hemp : s.nonempty) (hdir : s) (hbelow : ∀ (x : α), x sx < k) :
< 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} {f : β → α} (s : finset β) (h : ∀ (x : β), x s) :

Alias of the reverse direction of complete_lattice.well_founded_iff_is_Sup_finite_compact.

Alias of the reverse direction of complete_lattice.is_Sup_finite_compact_iff_is_sup_closed_compact.

theorem well_founded.is_sup_closed_compact (α : Type u_1)  :

Alias of the reverse direction of complete_lattice.is_sup_closed_compact_iff_well_founded.

theorem complete_lattice.well_founded.finite_of_independent {α : Type u_1} (hwf : well_founded gt) {ι : Type u_2} {t : ι → α} (ht : complete_lattice.independent t) (h_ne_bot : ∀ (i : ι), t i ) :
@[class]
structure is_compactly_generated (α : Type u_2)  :
Prop
• exists_Sup_eq : ∀ (x : α), ∃ (s : set α), (∀ (x : α), = x

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

Instances of this typeclass
@[simp]
theorem Sup_compact_le_eq {α : Type u_1} (b : α) :
has_Sup.Sup {c : α | = b
@[simp]
theorem Sup_compact_eq_top {α : Type u_1}  :
has_Sup.Sup {a : α | =
theorem le_iff_compact_le_imp {α : Type u_1} {a b : α} :
a b ∀ (c : α), c ac b
theorem inf_Sup_eq_of_directed_on {α : Type u_1} {a : α} {s : set α} (h : s) :
a = ⨆ (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} {a : α} {s : set α} :
a = ⨆ (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} {s : set α} :
∀ (t : finset α),
theorem complete_lattice.set_independent_Union_of_directed {α : Type u_1} {η : Type u_2} {s : η → set α} (hs : s) (h : ∀ (i : η), ) :
theorem complete_lattice.independent_sUnion_of_directed {α : Type u_1} {s : set (set α)} (hs : s) (h : ∀ (a : set α), ) :
theorem complete_lattice.Iic_coatomic_of_compact_element {α : Type u_1} {k : α}  :

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}  :
@[protected, instance]
def is_atomic_of_complemented_lattice {α : Type u_1}  :
@[protected, instance]
def is_atomistic_of_complemented_lattice {α : Type u_1}  :

See Lemma 5.1, Călugăreanu

theorem complemented_lattice_of_Sup_atoms_eq_top {α : Type u_1} (h : has_Sup.Sup {a : α | is_atom a} = ) :

See Theorem 6.6, Călugăreanu

theorem complemented_lattice_of_is_atomistic {α : Type u_1} [is_atomistic α] :

See Theorem 6.6, Călugăreanu

theorem complemented_lattice_iff_is_atomistic {α : Type u_1}  :