Graded orders

This file defines graded orders, also known as ranked orders.

An 𝕆-graded order is an order α equipped with a distinguished "grade" function α → 𝕆 which should be understood as giving the "height" of the elements. Usual graded orders are ℕ-graded, cograded orders are ℕᵒᵈ-graded, but we can also grade by ℤ, and polytopes are naturally Fin n-graded.

Visually, grade ℕ a is the height of a in the Hasse diagram of α.

Main declarations

• GradeOrder: Graded order.
• GradeMinOrder: Graded order where minimal elements have minimal grades.
• GradeMaxOrder: Graded order where maximal elements have maximal grades.
• GradeBoundedOrder: Graded order where minimal elements have minimal grades and maximal elements have maximal grades.
• grade: The grade of an element. Because an order can admit several gradings, the first argument is the order we grade by.

How to grade your order

Here are the translations between common references and our GradeOrder:

• [Stanley][stanley2012] defines a graded order of rank n as an order where all maximal chains have "length" n (so the number of elements of a chain is n + 1). This corresponds to GradeBoundedOrder (Fin (n + 1)) α.
• [Engel][engel1997]'s ranked orders are somewhere between GradeOrder ℕ α and GradeMinOrder ℕ α, in that he requires ∃ a, IsMin a ∧ grade ℕ a = 0 rather than ∀ a, IsMin a → grade ℕ a = 0. He defines a graded order as an order where all minimal elements have grade 0 and all maximal elements have the same grade. This is roughly a less bundled version of GradeBoundedOrder (Fin n) α, assuming we discard orders with infinite chains.

Implementation notes

One possible definition of graded orders is as the bounded orders whose flags (maximal chains) all have the same finite length (see Stanley p. 99). However, this means that all graded orders must have minimal and maximal elements and that the grade is not data.

Instead, we define graded orders by their grade function, without talking about flags yet.

References

• [Konrad Engel, Sperner Theory][engel1997]
• [Richard Stanley, Enumerative Combinatorics][stanley2012]

class GradeOrder (𝕆 : Type u_5) (α : Type u_6) [Preorder 𝕆] [Preorder α] : Type (max u_5 u_6)

An 𝕆-graded order is an order α equipped with a strictly monotone function grade 𝕆 : α → 𝕆 which preserves order covering (CovBy).

• grade : α → 𝕆

  The grading function.

• grade_strictMono : StrictMono GradeOrder.grade

  grade is strictly monotonic.

• covBy_grade ⦃a b : α⦄ : a ⋖ b → GradeOrder.grade a ⋖ GradeOrder.grade b

  grade preserves CovBy.

class GradeMinOrder (𝕆 : Type u_5) (α : Type u_6) [Preorder 𝕆] [Preorder α] extends GradeOrder 𝕆 α : Type (max u_5 u_6)

An 𝕆-graded order where minimal elements have minimal grades.

• grade : α → 𝕆
• grade_strictMono : StrictMono GradeOrder.grade
• covBy_grade ⦃a b : α⦄ : a ⋖ b → GradeOrder.grade a ⋖ GradeOrder.grade b
• isMin_grade ⦃a : α⦄ : IsMin a → IsMin (GradeOrder.grade a)

  Minimal elements have minimal grades.

class GradeMaxOrder (𝕆 : Type u_5) (α : Type u_6) [Preorder 𝕆] [Preorder α] extends GradeOrder 𝕆 α : Type (max u_5 u_6)

An 𝕆-graded order where maximal elements have maximal grades.

• grade : α → 𝕆
• grade_strictMono : StrictMono GradeOrder.grade
• covBy_grade ⦃a b : α⦄ : a ⋖ b → GradeOrder.grade a ⋖ GradeOrder.grade b
• isMax_grade ⦃a : α⦄ : IsMax a → IsMax (GradeOrder.grade a)

  Maximal elements have maximal grades.

class GradeBoundedOrder (𝕆 : Type u_5) (α : Type u_6) [Preorder 𝕆] [Preorder α] extends GradeMinOrder 𝕆 α, GradeMaxOrder 𝕆 α : Type (max u_5 u_6)

An 𝕆-graded order where minimal elements have minimal grades and maximal elements have maximal grades.

• grade : α → 𝕆
• grade_strictMono : StrictMono GradeOrder.grade
• covBy_grade ⦃a b : α⦄ : a ⋖ b → GradeOrder.grade a ⋖ GradeOrder.grade b
• isMin_grade ⦃a : α⦄ : IsMin a → IsMin (GradeOrder.grade a)
• isMax_grade ⦃a : α⦄ : IsMax a → IsMax (GradeOrder.grade a)

def grade (𝕆 : Type u_1) {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] : α → 𝕆

The grade of an element in a graded order. Morally, this is the number of elements you need to go down by to get to ⊥.

Equations

• grade 𝕆 = GradeOrder.grade

theorem CovBy.grade (𝕆 : Type u_1) {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] {a b : α} (h : a ⋖ b) : grade 𝕆 a ⋖ grade 𝕆 b

theorem grade_strictMono {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] : StrictMono (grade 𝕆)

theorem covBy_iff_lt_covBy_grade {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] {a b : α} : a ⋖ b ↔ a < b ∧ grade 𝕆 a ⋖ grade 𝕆 b

theorem IsMin.grade (𝕆 : Type u_1) {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMinOrder 𝕆 α] {a : α} (h : IsMin a) : IsMin (grade 𝕆 a)

@[simp]

theorem isMin_grade_iff {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMinOrder 𝕆 α] {a : α} : IsMin (grade 𝕆 a) ↔ IsMin a

theorem IsMax.grade (𝕆 : Type u_1) {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMaxOrder 𝕆 α] {a : α} (h : IsMax a) : IsMax (grade 𝕆 a)

@[simp]

theorem isMax_grade_iff {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMaxOrder 𝕆 α] {a : α} : IsMax (grade 𝕆 a) ↔ IsMax a

theorem grade_mono {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [PartialOrder α] [GradeOrder 𝕆 α] : Monotone (grade 𝕆)

theorem grade_injective {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] : Function.Injective (grade 𝕆)

@[simp]

theorem grade_le_grade_iff {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α} : grade 𝕆 a ≤ grade 𝕆 b ↔ a ≤ b

@[simp]

theorem grade_lt_grade_iff {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α} : grade 𝕆 a < grade 𝕆 b ↔ a < b

@[simp]

theorem grade_eq_grade_iff {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α} : grade 𝕆 a = grade 𝕆 b ↔ a = b

theorem grade_ne_grade_iff {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α} : grade 𝕆 a ≠ grade 𝕆 b ↔ a ≠ b

theorem grade_covBy_grade_iff {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α} : grade 𝕆 a ⋖ grade 𝕆 b ↔ a ⋖ b

@[simp]

theorem grade_bot {𝕆 : Type u_1} {α : Type u_3} [PartialOrder 𝕆] [Preorder α] [OrderBot 𝕆] [OrderBot α] [GradeMinOrder 𝕆 α] : grade 𝕆 ⊥ = ⊥

@[simp]

theorem grade_top {𝕆 : Type u_1} {α : Type u_3} [PartialOrder 𝕆] [Preorder α] [OrderTop 𝕆] [OrderTop α] [GradeMaxOrder 𝕆 α] : grade 𝕆 ⊤ = ⊤

Instances

@[implicit_reducible]

instance Preorder.toGradeBoundedOrder {α : Type u_3} [Preorder α] : GradeBoundedOrder α α

Equations

• Preorder.toGradeBoundedOrder = { grade := id, grade_strictMono := ⋯, covBy_grade := ⋯, isMin_grade := ⋯, isMax_grade := ⋯ }

@[simp]

theorem grade_self {α : Type u_3} [Preorder α] (a : α) : grade α a = a

Dual

@[implicit_reducible]

instance OrderDual.gradeOrder {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] : GradeOrder 𝕆ᵒᵈ αᵒᵈ

Equations

• OrderDual.gradeOrder = { grade := ⇑OrderDual.toDual ∘ grade 𝕆 ∘ ⇑OrderDual.ofDual, grade_strictMono := ⋯, covBy_grade := ⋯ }

@[implicit_reducible]

instance OrderDual.gradeMinOrder {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMaxOrder 𝕆 α] : GradeMinOrder 𝕆ᵒᵈ αᵒᵈ

Equations

• OrderDual.gradeMinOrder = { toGradeOrder := OrderDual.gradeOrder, isMin_grade := ⋯ }

@[implicit_reducible]

instance OrderDual.gradeMaxOrder {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMinOrder 𝕆 α] : GradeMaxOrder 𝕆ᵒᵈ αᵒᵈ

Equations

• OrderDual.gradeMaxOrder = { toGradeOrder := OrderDual.gradeOrder, isMax_grade := ⋯ }

@[implicit_reducible]

instance instGradeBoundedOrderOrderDual {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeBoundedOrder 𝕆 α] : GradeBoundedOrder 𝕆ᵒᵈ αᵒᵈ

Equations

• instGradeBoundedOrderOrderDual = { toGradeMinOrder := OrderDual.gradeMinOrder, isMax_grade := ⋯ }

@[simp]

theorem grade_toDual {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] (a : α) : grade 𝕆ᵒᵈ (OrderDual.toDual a) = OrderDual.toDual (grade 𝕆 a)

@[simp]

theorem grade_ofDual {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] (a : αᵒᵈ) : grade 𝕆 (OrderDual.ofDual a) = OrderDual.ofDual (grade 𝕆ᵒᵈ a)

Lifting a graded order

@[reducible, inline]

abbrev GradeOrder.liftLeft {𝕆 : Type u_1} {ℙ : Type u_2} {α : Type u_3} [Preorder 𝕆] [Preorder ℙ] [Preorder α] [GradeOrder 𝕆 α] (f : 𝕆 → ℙ) (hf : StrictMono f) (hcovBy : ∀ (a b : 𝕆), a ⋖ b → f a ⋖ f b) : GradeOrder ℙ α

Lifts a graded order along a strictly monotone function.

Equations

• GradeOrder.liftLeft f hf hcovBy = { grade := f ∘ grade 𝕆, grade_strictMono := ⋯, covBy_grade := ⋯ }

@[reducible, inline]

abbrev GradeMinOrder.liftLeft {𝕆 : Type u_1} {ℙ : Type u_2} {α : Type u_3} [Preorder 𝕆] [Preorder ℙ] [Preorder α] [GradeMinOrder 𝕆 α] (f : 𝕆 → ℙ) (hf : StrictMono f) (hcovBy : ∀ (a b : 𝕆), a ⋖ b → f a ⋖ f b) (hmin : ∀ (a : 𝕆), IsMin a → IsMin (f a)) : GradeMinOrder ℙ α

Lifts a graded order along a strictly monotone function.

Equations

• GradeMinOrder.liftLeft f hf hcovBy hmin = { toGradeOrder := GradeOrder.liftLeft f hf hcovBy, isMin_grade := ⋯ }

@[reducible, inline]

abbrev GradeMaxOrder.liftLeft {𝕆 : Type u_1} {ℙ : Type u_2} {α : Type u_3} [Preorder 𝕆] [Preorder ℙ] [Preorder α] [GradeMaxOrder 𝕆 α] (f : 𝕆 → ℙ) (hf : StrictMono f) (hcovBy : ∀ (a b : 𝕆), a ⋖ b → f a ⋖ f b) (hmax : ∀ (a : 𝕆), IsMax a → IsMax (f a)) : GradeMaxOrder ℙ α

Lifts a graded order along a strictly monotone function.

Equations

• GradeMaxOrder.liftLeft f hf hcovBy hmax = { toGradeOrder := GradeOrder.liftLeft f hf hcovBy, isMax_grade := ⋯ }

@[reducible, inline]

abbrev GradeBoundedOrder.liftLeft {𝕆 : Type u_1} {ℙ : Type u_2} {α : Type u_3} [Preorder 𝕆] [Preorder ℙ] [Preorder α] [GradeBoundedOrder 𝕆 α] (f : 𝕆 → ℙ) (hf : StrictMono f) (hcovBy : ∀ (a b : 𝕆), a ⋖ b → f a ⋖ f b) (hmin : ∀ (a : 𝕆), IsMin a → IsMin (f a)) (hmax : ∀ (a : 𝕆), IsMax a → IsMax (f a)) : GradeBoundedOrder ℙ α

Lifts a graded order along a strictly monotone function.

Equations

• GradeBoundedOrder.liftLeft f hf hcovBy hmin hmax = { toGradeMinOrder := GradeMinOrder.liftLeft f hf hcovBy hmin, isMax_grade := ⋯ }

@[reducible, inline]

abbrev GradeOrder.liftRight {𝕆 : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder 𝕆] [Preorder α] [Preorder β] [GradeOrder 𝕆 β] (f : α → β) (hf : StrictMono f) (hcovBy : ∀ (a b : α), a ⋖ b → f a ⋖ f b) : GradeOrder 𝕆 α

Lifts a graded order along a strictly monotone function.

Equations

• GradeOrder.liftRight f hf hcovBy = { grade := grade 𝕆 ∘ f, grade_strictMono := ⋯, covBy_grade := ⋯ }

@[reducible, inline]

abbrev GradeMinOrder.liftRight {𝕆 : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder 𝕆] [Preorder α] [Preorder β] [GradeMinOrder 𝕆 β] (f : α → β) (hf : StrictMono f) (hcovBy : ∀ (a b : α), a ⋖ b → f a ⋖ f b) (hmin : ∀ (a : α), IsMin a → IsMin (f a)) : GradeMinOrder 𝕆 α

Lifts a graded order along a strictly monotone function.

Equations

• GradeMinOrder.liftRight f hf hcovBy hmin = { toGradeOrder := GradeOrder.liftRight f hf hcovBy, isMin_grade := ⋯ }

@[reducible, inline]

abbrev GradeMaxOrder.liftRight {𝕆 : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder 𝕆] [Preorder α] [Preorder β] [GradeMaxOrder 𝕆 β] (f : α → β) (hf : StrictMono f) (hcovBy : ∀ (a b : α), a ⋖ b → f a ⋖ f b) (hmax : ∀ (a : α), IsMax a → IsMax (f a)) : GradeMaxOrder 𝕆 α

Lifts a graded order along a strictly monotone function.

Equations

• GradeMaxOrder.liftRight f hf hcovBy hmax = { toGradeOrder := GradeOrder.liftRight f hf hcovBy, isMax_grade := ⋯ }

@[reducible, inline]

abbrev GradeBoundedOrder.liftRight {𝕆 : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder 𝕆] [Preorder α] [Preorder β] [GradeBoundedOrder 𝕆 β] (f : α → β) (hf : StrictMono f) (hcovBy : ∀ (a b : α), a ⋖ b → f a ⋖ f b) (hmin : ∀ (a : α), IsMin a → IsMin (f a)) (hmax : ∀ (a : α), IsMax a → IsMax (f a)) : GradeBoundedOrder 𝕆 α

Lifts a graded order along a strictly monotone function.

Equations

• GradeBoundedOrder.liftRight f hf hcovBy hmin hmax = { toGradeMinOrder := GradeMinOrder.liftRight f hf hcovBy hmin, isMax_grade := ⋯ }

Fin n-graded to ℕ-graded to ℤ-graded

@[reducible, inline]

abbrev GradeOrder.finToNat {α : Type u_3} [Preorder α] (n : ℕ) [GradeOrder (Fin n) α] : GradeOrder ℕ α

A Fin n-graded order is also ℕ-graded. We do not mark this an instance because n is not inferable.

Equations

• GradeOrder.finToNat n = GradeOrder.liftLeft Fin.val ⋯ ⋯

@[reducible, inline]

abbrev GradeMinOrder.finToNat {α : Type u_3} [Preorder α] (n : ℕ) [GradeMinOrder (Fin n) α] : GradeMinOrder ℕ α

A Fin n-graded order is also ℕ-graded. We do not mark this an instance because n is not inferable.

Equations

• GradeMinOrder.finToNat n = GradeMinOrder.liftLeft Fin.val ⋯ ⋯ ⋯

@[implicit_reducible]

instance GradeOrder.natToInt {α : Type u_3} [Preorder α] [GradeOrder ℕ α] : GradeOrder ℤ α

Equations

• GradeOrder.natToInt = GradeOrder.liftLeft (fun (x : ℕ) => ↑x) Int.natCast_strictMono ⋯

theorem GradeOrder.wellFoundedLT {α : Type u_3} [Preorder α] (𝕆 : Type u_5) [Preorder 𝕆] [GradeOrder 𝕆 α] [WellFoundedLT 𝕆] : WellFoundedLT α

theorem GradeOrder.wellFoundedGT {α : Type u_3} [Preorder α] (𝕆 : Type u_5) [Preorder 𝕆] [GradeOrder 𝕆 α] [WellFoundedGT 𝕆] : WellFoundedGT α

instance instWellFoundedLTOfGradeOrderNat {α : Type u_3} [Preorder α] [GradeOrder ℕ α] : WellFoundedLT α

instance instWellFoundedGTOfGradeOrderOrderDualNat {α : Type u_3} [Preorder α] [GradeOrder ℕᵒᵈ α] : WellFoundedGT α

Grading a flag

A flag inherits the grading of its ambient order.

@[simp]

theorem Flag.coe_wcovBy_coe {α : Type u_3} [PartialOrder α] {s : Flag α} {a b : ↥s} : ↑a ⩿ ↑b ↔ a ⩿ b

@[simp]

theorem Flag.coe_covBy_coe {α : Type u_3} [PartialOrder α] {s : Flag α} {a b : ↥s} : ↑a ⋖ ↑b ↔ a ⋖ b

@[simp]

theorem Flag.isMax_coe {α : Type u_3} [PartialOrder α] {s : Flag α} {a : ↥s} : IsMax ↑a ↔ IsMax a

@[simp]

theorem Flag.isMin_coe {α : Type u_3} [PartialOrder α] {s : Flag α} {a : ↥s} : IsMin ↑a ↔ IsMin a

@[implicit_reducible]

instance Flag.instGradeOrderSubtypeMem {𝕆 : Type u_1} {α : Type u_3} [PartialOrder α] [Preorder 𝕆] [GradeOrder 𝕆 α] (s : Flag α) : GradeOrder 𝕆 ↥s

Equations

• s.instGradeOrderSubtypeMem = GradeOrder.liftRight Subtype.val ⋯ ⋯

@[implicit_reducible]

instance Flag.instGradeMinOrderSubtypeMem {𝕆 : Type u_1} {α : Type u_3} [PartialOrder α] [Preorder 𝕆] [GradeMinOrder 𝕆 α] (s : Flag α) : GradeMinOrder 𝕆 ↥s

Equations

• s.instGradeMinOrderSubtypeMem = GradeMinOrder.liftRight Subtype.val ⋯ ⋯ ⋯

@[implicit_reducible]

instance Flag.instGradeMaxOrderSubtypeMem {𝕆 : Type u_1} {α : Type u_3} [PartialOrder α] [Preorder 𝕆] [GradeMaxOrder 𝕆 α] (s : Flag α) : GradeMaxOrder 𝕆 ↥s

Equations

• s.instGradeMaxOrderSubtypeMem = GradeMaxOrder.liftRight Subtype.val ⋯ ⋯ ⋯

@[implicit_reducible]

instance Flag.instGradeBoundedOrderSubtypeMem {𝕆 : Type u_1} {α : Type u_3} [PartialOrder α] [Preorder 𝕆] [GradeBoundedOrder 𝕆 α] (s : Flag α) : GradeBoundedOrder 𝕆 ↥s

Equations

• s.instGradeBoundedOrderSubtypeMem = GradeBoundedOrder.liftRight Subtype.val ⋯ ⋯ ⋯ ⋯

@[simp]

theorem Flag.grade_coe {𝕆 : Type u_1} {α : Type u_3} [PartialOrder α] {s : Flag α} [Preorder 𝕆] [GradeOrder 𝕆 α] (a : ↥s) : grade 𝕆 ↑a = grade 𝕆 a