Documentation

Mathlib.Order.Hom.Order

Lattice structure on order homomorphisms #

This file defines the lattice structure on order homomorphisms, which are bundled monotone functions.

Main definitions #

Tags #

monotone map, bundled morphism

Equations
  • OrderHom.instSupOrderHomToPreorderToPartialOrder = { sup := fun (f g : α →o β) => { toFun := fun (a : α) => f a g a, monotone' := (_ : Monotone (f fun (a : α) => g a)) } }
@[simp]
theorem OrderHom.coe_sup {α : Type u_1} {β : Type u_2} [Preorder α] [SemilatticeSup β] (f : α →o β) (g : α →o β) :
(f g) = f g
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • OrderHom.instInfOrderHomToPreorderToPartialOrder = { inf := fun (f g : α →o β) => { toFun := fun (a : α) => f a g a, monotone' := (_ : Monotone (f fun (a : α) => g a)) } }
@[simp]
theorem OrderHom.coe_inf {α : Type u_1} {β : Type u_2} [Preorder α] [SemilatticeInf β] (f : α →o β) (g : α →o β) :
(f g) = f g
Equations
  • One or more equations did not get rendered due to their size.
instance OrderHom.lattice {α : Type u_1} {β : Type u_2} [Preorder α] [Lattice β] :
Lattice (α →o β)
Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem OrderHom.instBotOrderHom_bot {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [OrderBot β] :
instance OrderHom.instBotOrderHom {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [OrderBot β] :
Bot (α →o β)
Equations
instance OrderHom.orderBot {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [OrderBot β] :
OrderBot (α →o β)
Equations
@[simp]
theorem OrderHom.instTopOrderHom_top {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [OrderTop β] :
instance OrderHom.instTopOrderHom {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [OrderTop β] :
Top (α →o β)
Equations
instance OrderHom.orderTop {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [OrderTop β] :
OrderTop (α →o β)
Equations
Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem OrderHom.sInf_apply {α : Type u_1} {β : Type u_2} [Preorder α] [CompleteLattice β] (s : Set (α →o β)) (x : α) :
(sInf s) x = ⨅ f ∈ s, f x
theorem OrderHom.iInf_apply {α : Type u_1} {β : Type u_2} [Preorder α] {ι : Sort u_3} [CompleteLattice β] (f : ια →o β) (x : α) :
(⨅ (i : ι), f i) x = ⨅ (i : ι), (f i) x
@[simp]
theorem OrderHom.coe_iInf {α : Type u_1} {β : Type u_2} [Preorder α] {ι : Sort u_3} [CompleteLattice β] (f : ια →o β) :
(⨅ (i : ι), f i) = ⨅ (i : ι), (f i)
Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem OrderHom.sSup_apply {α : Type u_1} {β : Type u_2} [Preorder α] [CompleteLattice β] (s : Set (α →o β)) (x : α) :
(sSup s) x = ⨆ f ∈ s, f x
theorem OrderHom.iSup_apply {α : Type u_1} {β : Type u_2} [Preorder α] {ι : Sort u_3} [CompleteLattice β] (f : ια →o β) (x : α) :
(⨆ (i : ι), f i) x = ⨆ (i : ι), (f i) x
@[simp]
theorem OrderHom.coe_iSup {α : Type u_1} {β : Type u_2} [Preorder α] {ι : Sort u_3} [CompleteLattice β] (f : ια →o β) :
(⨆ (i : ι), f i) = ⨆ (i : ι), (f i)
Equations
  • One or more equations did not get rendered due to their size.
theorem OrderHom.iterate_sup_le_sup_iff {α : Type u_3} [SemilatticeSup α] (f : α →o α) :
(∀ (n₁ n₂ : ) (a₁ a₂ : α), (f)^[n₁ + n₂] (a₁ a₂) (f)^[n₁] a₁ (f)^[n₂] a₂) ∀ (a₁ a₂ : α), f (a₁ a₂) f a₁ a₂