Documentation

Mathlib.Order.ModularLattice

Modular Lattices #

This file defines (semi)modular lattices, a kind of lattice useful in algebra. For examples, look to the subobject lattices of abelian groups, submodules, and ideals, or consider any distributive lattice.

Typeclasses #

We define (semi)modularity typeclasses as Prop-valued mixins.

Main Definitions #

Main Results #

References #

TODO #

  • covby_sup_of_inf_covby_covby : ∀ {a b : α}, a b aa b ba a b

    a ⊔ b covers a and b if a and b both cover a ⊓ b.

A weakly upper modular lattice is a lattice where a ⊔ b covers a and b if a and b both cover a ⊓ b.

Instances
    • inf_covby_of_covby_covby_sup : ∀ {a b : α}, a a bb a ba b a

      a and b cover a ⊓ b if a ⊔ b covers both a and b

    A weakly lower modular lattice is a lattice where a and b cover a ⊓ b if a ⊔ b covers both a and b.

    Instances
      class IsUpperModularLattice (α : Type u_2) [Lattice α] :
      • covby_sup_of_inf_covby : ∀ {a b : α}, a b ab a b

        a ⊔ b covers a and b if either a or b covers a ⊓ b

      An upper modular lattice, aka semimodular lattice, is a lattice where a ⊔ b covers a and b if either a or b covers a ⊓ b.

      Instances
        class IsLowerModularLattice (α : Type u_2) [Lattice α] :
        • inf_covby_of_covby_sup : ∀ {a b : α}, a a ba b b

          a and b both cover a ⊓ b if a ⊔ b covers either a or b

        A lower modular lattice is a lattice where a and b both cover a ⊓ b if a ⊔ b covers either a or b.

        Instances
          class IsModularLattice (α : Type u_2) [Lattice α] :
          • sup_inf_le_assoc_of_le : ∀ {x : α} (y : α) {z : α}, x z(x y) z x y z

            Whenever x ≤ z, then for any y, (x ⊔ y) ⊓ z ≤ x ⊔ (y ⊓ z)

          A modular lattice is one with a limited associativity between and .

          Instances
            theorem covby_sup_of_inf_covby_of_inf_covby_left {α : Type u_1} [Lattice α] [IsWeakUpperModularLattice α] {a : α} {b : α} :
            a b aa b ba a b
            theorem covby_sup_of_inf_covby_of_inf_covby_right {α : Type u_1} [Lattice α] [IsWeakUpperModularLattice α] {a : α} {b : α} :
            a b aa b bb a b
            theorem Covby.sup_of_inf_of_inf_left {α : Type u_1} [Lattice α] [IsWeakUpperModularLattice α] {a : α} {b : α} :
            a b aa b ba a b

            Alias of covby_sup_of_inf_covby_of_inf_covby_left.

            theorem Covby.sup_of_inf_of_inf_right {α : Type u_1} [Lattice α] [IsWeakUpperModularLattice α] {a : α} {b : α} :
            a b aa b bb a b

            Alias of covby_sup_of_inf_covby_of_inf_covby_right.

            theorem inf_covby_of_covby_sup_of_covby_sup_left {α : Type u_1} [Lattice α] [IsWeakLowerModularLattice α] {a : α} {b : α} :
            a a bb a ba b a
            theorem inf_covby_of_covby_sup_of_covby_sup_right {α : Type u_1} [Lattice α] [IsWeakLowerModularLattice α] {a : α} {b : α} :
            a a bb a ba b b
            theorem Covby.inf_of_sup_of_sup_left {α : Type u_1} [Lattice α] [IsWeakLowerModularLattice α] {a : α} {b : α} :
            a a bb a ba b a

            Alias of inf_covby_of_covby_sup_of_covby_sup_left.

            theorem Covby.inf_of_sup_of_sup_right {α : Type u_1} [Lattice α] [IsWeakLowerModularLattice α] {a : α} {b : α} :
            a a bb a ba b b

            Alias of inf_covby_of_covby_sup_of_covby_sup_right.

            theorem covby_sup_of_inf_covby_left {α : Type u_1} [Lattice α] [IsUpperModularLattice α] {a : α} {b : α} :
            a b ab a b
            theorem covby_sup_of_inf_covby_right {α : Type u_1} [Lattice α] [IsUpperModularLattice α] {a : α} {b : α} :
            a b ba a b
            theorem Covby.sup_of_inf_left {α : Type u_1} [Lattice α] [IsUpperModularLattice α] {a : α} {b : α} :
            a b ab a b

            Alias of covby_sup_of_inf_covby_left.

            theorem Covby.sup_of_inf_right {α : Type u_1} [Lattice α] [IsUpperModularLattice α] {a : α} {b : α} :
            a b ba a b

            Alias of covby_sup_of_inf_covby_right.

            theorem inf_covby_of_covby_sup_left {α : Type u_1} [Lattice α] [IsLowerModularLattice α] {a : α} {b : α} :
            a a ba b b
            theorem inf_covby_of_covby_sup_right {α : Type u_1} [Lattice α] [IsLowerModularLattice α] {a : α} {b : α} :
            b a ba b a
            theorem Covby.inf_of_sup_left {α : Type u_1} [Lattice α] [IsLowerModularLattice α] {a : α} {b : α} :
            a a ba b b

            Alias of inf_covby_of_covby_sup_left.

            theorem Covby.inf_of_sup_right {α : Type u_1} [Lattice α] [IsLowerModularLattice α] {a : α} {b : α} :
            b a ba b a

            Alias of inf_covby_of_covby_sup_right.

            theorem sup_inf_assoc_of_le {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} (y : α) {z : α} (h : x z) :
            (x y) z = x y z
            theorem IsModularLattice.inf_sup_inf_assoc {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} {y : α} {z : α} :
            x z y z = (x z y) z
            theorem inf_sup_assoc_of_le {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} (y : α) {z : α} (h : z x) :
            x y z = x (y z)
            theorem IsModularLattice.sup_inf_sup_assoc {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} {y : α} {z : α} :
            (x z) (y z) = (x z) y z
            theorem eq_of_le_of_inf_le_of_sup_le {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} {y : α} {z : α} (hxy : x y) (hinf : y z x z) (hsup : y z x z) :
            x = y
            theorem sup_lt_sup_of_lt_of_inf_le_inf {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} {y : α} {z : α} (hxy : x < y) (hinf : y z x z) :
            x z < y z
            theorem inf_lt_inf_of_lt_of_sup_le_sup {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} {y : α} {z : α} (hxy : x < y) (hinf : y z x z) :
            x z < y z
            theorem wellFounded_lt_exact_sequence {α : Type u_1} [Lattice α] [IsModularLattice α] {β : Type u_2} {γ : Type u_3} [PartialOrder β] [Preorder γ] (h₁ : WellFounded fun x x_1 => x < x_1) (h₂ : WellFounded fun x x_1 => x < x_1) (K : α) (f₁ : βα) (f₂ : αβ) (g₁ : γα) (g₂ : αγ) (gci : GaloisCoinsertion f₁ f₂) (gi : GaloisInsertion g₂ g₁) (hf : ∀ (a : α), f₁ (f₂ a) = a K) (hg : ∀ (a : α), g₁ (g₂ a) = a K) :
            WellFounded fun x x_1 => x < x_1

            A generalization of the theorem that if N is a submodule of M and N and M / N are both Artinian, then M is Artinian.

            theorem wellFounded_gt_exact_sequence {α : Type u_1} [Lattice α] [IsModularLattice α] {β : Type u_2} {γ : Type u_3} [Preorder β] [PartialOrder γ] (h₁ : WellFounded fun x x_1 => x > x_1) (h₂ : WellFounded fun x x_1 => x > x_1) (K : α) (f₁ : βα) (f₂ : αβ) (g₁ : γα) (g₂ : αγ) (gci : GaloisCoinsertion f₁ f₂) (gi : GaloisInsertion g₂ g₁) (hf : ∀ (a : α), f₁ (f₂ a) = a K) (hg : ∀ (a : α), g₁ (g₂ a) = a K) :
            WellFounded fun x x_1 => x > x_1

            A generalization of the theorem that if N is a submodule of M and N and M / N are both Noetherian, then M is Noetherian.

            @[simp]
            theorem infIccOrderIsoIccSup_apply_coe {α : Type u_1} [Lattice α] [IsModularLattice α] (a : α) (b : α) (x : ↑(Set.Icc (a b) a)) :
            ↑(↑(infIccOrderIsoIccSup a b) x) = x b
            @[simp]
            theorem infIccOrderIsoIccSup_symm_apply_coe {α : Type u_1} [Lattice α] [IsModularLattice α] (a : α) (b : α) (x : ↑(Set.Icc b (a b))) :
            ↑(↑(RelIso.symm (infIccOrderIsoIccSup a b)) x) = a x
            def infIccOrderIsoIccSup {α : Type u_1} [Lattice α] [IsModularLattice α] (a : α) (b : α) :
            ↑(Set.Icc (a b) a) ≃o ↑(Set.Icc b (a b))

            The diamond isomorphism between the intervals [a ⊓ b, a] and [b, a ⊔ b]

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
              theorem inf_strictMonoOn_Icc_sup {α : Type u_1} [Lattice α] [IsModularLattice α] {a : α} {b : α} :
              StrictMonoOn (fun c => a c) (Set.Icc b (a b))
              theorem sup_strictMonoOn_Icc_inf {α : Type u_1} [Lattice α] [IsModularLattice α] {a : α} {b : α} :
              StrictMonoOn (fun c => c b) (Set.Icc (a b) a)
              @[simp]
              theorem infIooOrderIsoIooSup_apply_coe {α : Type u_1} [Lattice α] [IsModularLattice α] (a : α) (b : α) (c : ↑(Set.Ioo (a b) a)) :
              ↑(↑(infIooOrderIsoIooSup a b) c) = c b
              @[simp]
              theorem infIooOrderIsoIooSup_symm_apply_coe {α : Type u_1} [Lattice α] [IsModularLattice α] (a : α) (b : α) (c : ↑(Set.Ioo b (a b))) :
              ↑(↑(RelIso.symm (infIooOrderIsoIooSup a b)) c) = a c
              def infIooOrderIsoIooSup {α : Type u_1} [Lattice α] [IsModularLattice α] (a : α) (b : α) :
              ↑(Set.Ioo (a b) a) ≃o ↑(Set.Ioo b (a b))

              The diamond isomorphism between the intervals ]a ⊓ b, a[ and }b, a ⊔ b[.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                def IsCompl.IicOrderIsoIci {α : Type u_1} [Lattice α] [BoundedOrder α] [IsModularLattice α] {a : α} {b : α} (h : IsCompl a b) :
                ↑(Set.Iic a) ≃o ↑(Set.Ici b)

                The diamond isomorphism between the intervals Set.Iic a and Set.Ici b.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For
                  theorem isModularLattice_iff_inf_sup_inf_assoc {α : Type u_1} [Lattice α] :
                  IsModularLattice α ∀ (x y z : α), x z y z = (x z y) z
                  theorem Disjoint.disjoint_sup_right_of_disjoint_sup_left {α : Type u_1} [Lattice α] [OrderBot α] [IsModularLattice α] {a : α} {b : α} {c : α} (h : Disjoint a b) (hsup : Disjoint (a b) c) :
                  Disjoint a (b c)
                  theorem Disjoint.disjoint_sup_left_of_disjoint_sup_right {α : Type u_1} [Lattice α] [OrderBot α] [IsModularLattice α] {a : α} {b : α} {c : α} (h : Disjoint b c) (hsup : Disjoint a (b c)) :
                  Disjoint (a b) c