Atoms, Coatoms, Simple Lattices, and Finiteness

This module contains some results on atoms and simple lattices in the finite context.

Main results

• Finite.to_isAtomic, Finite.to_isCoatomic: Finite partial orders with bottom resp. top are atomic resp. coatomic.

@[instance 200]

instance IsSimpleOrder.instFintypeOfDecidableEq {α : Type u_1} [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α] : Fintype α

Equations

• IsSimpleOrder.instFintypeOfDecidableEq = Fintype.ofEquiv Bool IsSimpleOrder.equivBool.symm

@[instance 200]

instance IsSimpleOrder.instFinite {α : Type u_1} [LE α] [BoundedOrder α] [IsSimpleOrder α] : Finite α

Equations

• ⋯ = ⋯

theorem Fintype.IsSimpleOrder.univ {α : Type u_1} [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α] : Finset.univ = {⊤, ⊥}

theorem Fintype.IsSimpleOrder.card {α : Type u_1} [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α] : Fintype.card α = 2

instance Bool.instIsSimpleOrder : IsSimpleOrder Bool

Equations

• Bool.instIsSimpleOrder = ⋯

@[instance 100]

instance Finite.to_isCoatomic {α : Type u_1} [PartialOrder α] [OrderTop α] [Finite α] : IsCoatomic α

Equations

• ⋯ = ⋯

@[instance 100]

instance Finite.to_isAtomic {α : Type u_1} [PartialOrder α] [OrderBot α] [Finite α] : IsAtomic α

Equations

• ⋯ = ⋯

instance instIsStronglyAtomic {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] : IsStronglyAtomic α

Equations

• ⋯ = ⋯

instance instIsStronglyCoatomic {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] : IsStronglyCoatomic α

Equations

• ⋯ = ⋯

theorem exists_covby_infinite_Ici_of_infinite_Ici {α : Type u_1} [PartialOrder α] {a : α} [IsStronglyAtomic α] (ha : (Set.Ici a).Infinite) (hfin : {x : α | a ⋖ x}.Finite) : ∃ (b : α), a ⋖ b ∧ (Set.Ici b).Infinite

theorem exists_covby_infinite_Iic_of_infinite_Iic {α : Type u_1} [PartialOrder α] {a : α} [IsStronglyCoatomic α] (ha : (Set.Iic a).Infinite) (hfin : {x : α | x ⋖ a}.Finite) : ∃ (b : α), b ⋖ a ∧ (Set.Iic b).Infinite