Documentation

Mathlib.Order.Closure

Closure operators between preorders #

We define (bundled) closure operators on a preorder as monotone (increasing), extensive (inflationary) and idempotent functions. We define closed elements for the operator as elements which are fixed by it.

Lower adjoints to a function between preorders u : β → α allow to generalise closure operators to situations where the closure operator we are dealing with naturally decomposes as u ∘ l where l is a worthy function to have on its own. Typical examples include l : Set G → Subgroup G := Subgroup.closure, u : Subgroup G → Set G := (↑), where G is a group. This shows there is a close connection between closure operators, lower adjoints and Galois connections/insertions: every Galois connection induces a lower adjoint which itself induces a closure operator by composition (see GaloisConnection.lowerAdjoint and LowerAdjoint.closureOperator), and every closure operator on a partial order induces a Galois insertion from the set of closed elements to the underlying type (see ClosureOperator.gi).

Main definitions #

Implementation details #

Although LowerAdjoint is technically a generalisation of ClosureOperator (by defining toFun := id), it is desirable to have both as otherwise ids would be carried all over the place when using concrete closure operators such as ConvexHull.

LowerAdjoint really is a semibundled structure version of GaloisConnection.

References #

Closure operator #

structure ClosureOperator (α : Type u_1) [Preorder α] extends OrderHom :
Type u_1

A closure operator on the preorder α is a monotone function which is extensive (every x is less than its closure) and idempotent.

  • toFun : αα
  • monotone' : Monotone self.toFun
  • le_closure' : ∀ (x : α), x self.toFun x

    An element is less than or equal its closure

  • idempotent' : ∀ (x : α), self.toFun (self.toFun x) = self.toFun x

    Closures are idempotent

  • IsClosed : αProp

    Predicate for an element to be closed.

    By default, this is defined as c.IsClosed x := (c x = x) (see isClosed_iff). We allow an override to fix definitional equalities.

  • isClosed_iff : ∀ {x : α}, self.IsClosed x self.toFun x = x
Instances For
    Equations
    @[simp]
    theorem ClosureOperator.id_isClosed (α : Type u_1) [PartialOrder α] :
    ∀ (x : α), (ClosureOperator.id α).IsClosed x = True
    @[simp]
    theorem ClosureOperator.id_apply (α : Type u_1) [PartialOrder α] (a : α) :

    The identity function as a closure operator.

    Equations
    • ClosureOperator.id α = { toOrderHom := OrderHom.id, le_closure' := , idempotent' := , IsClosed := fun (x : α) => True, isClosed_iff := }
    Instances For
      theorem ClosureOperator.ext {α : Type u_1} [PartialOrder α] (c₁ : ClosureOperator α) (c₂ : ClosureOperator α) :
      c₁ = c₂c₁ = c₂
      @[simp]
      theorem ClosureOperator.mk'_isClosed {α : Type u_1} [PartialOrder α] (f : αα) (hf₁ : Monotone f) (hf₂ : ∀ (x : α), x f x) (hf₃ : ∀ (x : α), f (f x) f x) (x : α) :
      (ClosureOperator.mk' f hf₁ hf₂ hf₃).IsClosed x = (f x = x)
      @[simp]
      theorem ClosureOperator.mk'_apply {α : Type u_1} [PartialOrder α] (f : αα) (hf₁ : Monotone f) (hf₂ : ∀ (x : α), x f x) (hf₃ : ∀ (x : α), f (f x) f x) :
      ∀ (a : α), (ClosureOperator.mk' f hf₁ hf₂ hf₃) a = f a
      def ClosureOperator.mk' {α : Type u_1} [PartialOrder α] (f : αα) (hf₁ : Monotone f) (hf₂ : ∀ (x : α), x f x) (hf₃ : ∀ (x : α), f (f x) f x) :

      Constructor for a closure operator using the weaker idempotency axiom: f (f x) ≤ f x.

      Equations
      • ClosureOperator.mk' f hf₁ hf₂ hf₃ = { toOrderHom := { toFun := f, monotone' := hf₁ }, le_closure' := hf₂, idempotent' := , IsClosed := fun (x : α) => f x = x, isClosed_iff := }
      Instances For
        @[simp]
        theorem ClosureOperator.mk₂_apply {α : Type u_1} [PartialOrder α] (f : αα) (hf : ∀ (x : α), x f x) (hmin : ∀ ⦃x y : α⦄, x f yf x f y) :
        ∀ (a : α), (ClosureOperator.mk₂ f hf hmin) a = f a
        @[simp]
        theorem ClosureOperator.mk₂_isClosed {α : Type u_1} [PartialOrder α] (f : αα) (hf : ∀ (x : α), x f x) (hmin : ∀ ⦃x y : α⦄, x f yf x f y) (x : α) :
        (ClosureOperator.mk₂ f hf hmin).IsClosed x = (f x = x)
        def ClosureOperator.mk₂ {α : Type u_1} [PartialOrder α] (f : αα) (hf : ∀ (x : α), x f x) (hmin : ∀ ⦃x y : α⦄, x f yf x f y) :

        Convenience constructor for a closure operator using the weaker minimality axiom: x ≤ f y → f x ≤ f y, which is sometimes easier to prove in practice.

        Equations
        • ClosureOperator.mk₂ f hf hmin = { toOrderHom := { toFun := f, monotone' := }, le_closure' := hf, idempotent' := , IsClosed := fun (x : α) => f x = x, isClosed_iff := }
        Instances For
          @[simp]
          theorem ClosureOperator.ofPred_isClosed {α : Type u_1} [PartialOrder α] (f : αα) (p : αProp) (hf : ∀ (x : α), x f x) (hfp : ∀ (x : α), p (f x)) (hmin : ∀ ⦃x y : α⦄, x yp yf x y) :
          ∀ (a : α), (ClosureOperator.ofPred f p hf hfp hmin).IsClosed a = p a
          @[simp]
          theorem ClosureOperator.ofPred_apply {α : Type u_1} [PartialOrder α] (f : αα) (p : αProp) (hf : ∀ (x : α), x f x) (hfp : ∀ (x : α), p (f x)) (hmin : ∀ ⦃x y : α⦄, x yp yf x y) :
          ∀ (a : α), (ClosureOperator.ofPred f p hf hfp hmin) a = f a
          def ClosureOperator.ofPred {α : Type u_1} [PartialOrder α] (f : αα) (p : αProp) (hf : ∀ (x : α), x f x) (hfp : ∀ (x : α), p (f x)) (hmin : ∀ ⦃x y : α⦄, x yp yf x y) :

          Construct a closure operator from an inflationary function f and a "closedness" predicate p witnessing minimality of f x among closed elements greater than x.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
            theorem ClosureOperator.le_closure {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) (x : α) :
            x c x

            Every element is less than its closure. This property is sometimes referred to as extensivity or inflationarity.

            @[simp]
            theorem ClosureOperator.idempotent {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) (x : α) :
            c (c x) = c x
            @[simp]
            theorem ClosureOperator.isClosed_closure {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) (x : α) :
            c.IsClosed (c x)
            @[inline, reducible]
            abbrev ClosureOperator.Closeds {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) :
            Type u_1

            The type of elements closed under a closure operator.

            Equations
            Instances For

              Send an element to a closed element (by taking the closure).

              Equations
              Instances For
                theorem ClosureOperator.IsClosed.closure_eq {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} :
                c.IsClosed xc x = x
                theorem ClosureOperator.isClosed_iff_closure_le {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} :
                c.IsClosed x c x x
                theorem ClosureOperator.setOf_isClosed_eq_range_closure {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} :
                {x : α | c.IsClosed x} = Set.range c

                The set of closed elements for c is exactly its range.

                theorem ClosureOperator.le_closure_iff {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} {y : α} :
                x c y c x c y
                @[simp]
                theorem ClosureOperator.IsClosed.closure_le_iff {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} {y : α} (hy : c.IsClosed y) :
                c x y x y
                theorem ClosureOperator.closure_min {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} {y : α} (hxy : x y) (hy : c.IsClosed y) :
                c x y
                theorem ClosureOperator.eq_ofPred_closed {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) :
                c = ClosureOperator.ofPred (c) c.IsClosed

                A closure operator is equal to the closure operator obtained by feeding c.closed into the ofPred constructor.

                @[simp]
                @[simp]
                theorem ClosureOperator.isClosed_top {α : Type u_1} [PartialOrder α] [OrderTop α] (c : ClosureOperator α) :
                c.IsClosed
                theorem ClosureOperator.closure_inf_le {α : Type u_1} [SemilatticeInf α] (c : ClosureOperator α) (x : α) (y : α) :
                c (x y) c x c y
                theorem ClosureOperator.closure_sup_closure_le {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                c x c y c (x y)
                theorem ClosureOperator.closure_sup_closure_left {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                c (c x y) = c (x y)
                theorem ClosureOperator.closure_sup_closure_right {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                c (x c y) = c (x y)
                theorem ClosureOperator.closure_sup_closure {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                c (c x c y) = c (x y)
                @[simp]
                theorem ClosureOperator.ofCompletePred_isClosed {α : Type u_1} [CompleteLattice α] (p : αProp) (hsinf : ∀ (s : Set α), (as, p a)p (sInf s)) :
                ∀ (a : α), (ClosureOperator.ofCompletePred p hsinf).IsClosed a = p a
                @[simp]
                theorem ClosureOperator.ofCompletePred_apply {α : Type u_1} [CompleteLattice α] (p : αProp) (hsinf : ∀ (s : Set α), (as, p a)p (sInf s)) (a : α) :
                (ClosureOperator.ofCompletePred p hsinf) a = ⨅ (b : { b : α // a b p b }), b
                def ClosureOperator.ofCompletePred {α : Type u_1} [CompleteLattice α] (p : αProp) (hsinf : ∀ (s : Set α), (as, p a)p (sInf s)) :

                Define a closure operator from a predicate that's preserved under infima.

                Equations
                Instances For
                  @[simp]
                  theorem ClosureOperator.closure_iSup_closure {α : Type u_1} {ι : Sort u_2} [CompleteLattice α] (c : ClosureOperator α) (f : ια) :
                  c (⨆ (i : ι), c (f i)) = c (⨆ (i : ι), f i)
                  @[simp]
                  theorem ClosureOperator.closure_iSup₂_closure {α : Type u_1} {ι : Sort u_2} {κ : ιSort u_3} [CompleteLattice α] (c : ClosureOperator α) (f : (i : ι) → κ iα) :
                  c (⨆ (i : ι), ⨆ (j : κ i), c (f i j)) = c (⨆ (i : ι), ⨆ (j : κ i), f i j)

                  Lower adjoint #

                  structure LowerAdjoint {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] (u : βα) :
                  Type (max u_1 u_4)

                  A lower adjoint of u on the preorder α is a function l such that l and u form a Galois connection. It allows us to define closure operators whose output does not match the input. In practice, u is often (↑) : β → α.

                  • toFun : αβ

                    The underlying function

                  • gc' : GaloisConnection self.toFun u

                    The underlying function is a lower adjoint.

                  Instances For
                    @[simp]
                    theorem LowerAdjoint.id_toFun (α : Type u_1) [Preorder α] (x : α) :
                    (LowerAdjoint.id α).toFun x = x
                    def LowerAdjoint.id (α : Type u_1) [Preorder α] :

                    The identity function as a lower adjoint to itself.

                    Equations
                    Instances For
                      Equations
                      instance LowerAdjoint.instCoeFunLowerAdjointForAll {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} :
                      CoeFun (LowerAdjoint u) fun (x : LowerAdjoint u) => αβ
                      Equations
                      • LowerAdjoint.instCoeFunLowerAdjointForAll = { coe := LowerAdjoint.toFun }
                      theorem LowerAdjoint.gc {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                      theorem LowerAdjoint.ext {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l₁ : LowerAdjoint u) (l₂ : LowerAdjoint u) :
                      l₁.toFun = l₂.toFunl₁ = l₂
                      theorem LowerAdjoint.monotone {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                      Monotone (u l.toFun)
                      theorem LowerAdjoint.le_closure {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                      x u (l.toFun x)

                      Every element is less than its closure. This property is sometimes referred to as extensivity or inflationarity.

                      @[simp]
                      theorem LowerAdjoint.closureOperator_apply {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                      (LowerAdjoint.closureOperator l) x = u (l.toFun x)
                      @[simp]
                      theorem LowerAdjoint.closureOperator_isClosed {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                      (LowerAdjoint.closureOperator l).IsClosed x = (u (l.toFun x) = x)
                      def LowerAdjoint.closureOperator {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :

                      Every lower adjoint induces a closure operator given by the composition. This is the partial order version of the statement that every adjunction induces a monad.

                      Equations
                      • One or more equations did not get rendered due to their size.
                      Instances For
                        theorem LowerAdjoint.idempotent {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                        u (l.toFun (u (l.toFun x))) = u (l.toFun x)
                        theorem LowerAdjoint.le_closure_iff {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                        x u (l.toFun y) u (l.toFun x) u (l.toFun y)
                        def LowerAdjoint.closed {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                        Set α

                        An element x is closed for l : LowerAdjoint u if it is a fixed point: u (l x) = x

                        Equations
                        Instances For
                          theorem LowerAdjoint.mem_closed_iff {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                          x LowerAdjoint.closed l u (l.toFun x) = x
                          theorem LowerAdjoint.closure_eq_self_of_mem_closed {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) {x : α} (h : x LowerAdjoint.closed l) :
                          u (l.toFun x) = x
                          theorem LowerAdjoint.mem_closed_iff_closure_le {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                          x LowerAdjoint.closed l u (l.toFun x) x
                          @[simp]
                          theorem LowerAdjoint.closure_is_closed {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                          u (l.toFun x) LowerAdjoint.closed l
                          theorem LowerAdjoint.closed_eq_range_close {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) :

                          The set of closed elements for l is the range of u ∘ l.

                          def LowerAdjoint.toClosed {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) :

                          Send an x to an element of the set of closed elements (by taking the closure).

                          Equations
                          Instances For
                            @[simp]
                            theorem LowerAdjoint.closure_le_closed_iff_le {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) {y : α} (hy : LowerAdjoint.closed l y) :
                            u (l.toFun x) y x y
                            theorem LowerAdjoint.closure_top {α : Type u_1} {β : Type u_4} [PartialOrder α] [OrderTop α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                            u (l.toFun ) =
                            theorem LowerAdjoint.closure_inf_le {α : Type u_1} {β : Type u_4} [SemilatticeInf α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                            u (l.toFun (x y)) u (l.toFun x) u (l.toFun y)
                            theorem LowerAdjoint.closure_sup_closure_le {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                            u (l.toFun x) u (l.toFun y) u (l.toFun (x y))
                            theorem LowerAdjoint.closure_sup_closure_left {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                            u (l.toFun (u (l.toFun x) y)) = u (l.toFun (x y))
                            theorem LowerAdjoint.closure_sup_closure_right {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                            u (l.toFun (x u (l.toFun y))) = u (l.toFun (x y))
                            theorem LowerAdjoint.closure_sup_closure {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                            u (l.toFun (u (l.toFun x) u (l.toFun y))) = u (l.toFun (x y))
                            theorem LowerAdjoint.closure_iSup_closure {α : Type u_1} {ι : Sort u_2} {β : Type u_4} [CompleteLattice α] [Preorder β] {u : βα} (l : LowerAdjoint u) (f : ια) :
                            u (l.toFun (⨆ (i : ι), u (l.toFun (f i)))) = u (l.toFun (⨆ (i : ι), f i))
                            theorem LowerAdjoint.closure_iSup₂_closure {α : Type u_1} {ι : Sort u_2} {κ : ιSort u_3} {β : Type u_4} [CompleteLattice α] [Preorder β] {u : βα} (l : LowerAdjoint u) (f : (i : ι) → κ iα) :
                            u (l.toFun (⨆ (i : ι), ⨆ (j : κ i), u (l.toFun (f i j)))) = u (l.toFun (⨆ (i : ι), ⨆ (j : κ i), f i j))
                            theorem LowerAdjoint.subset_closure {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (s : Set β) :
                            s (l.toFun s)
                            theorem LowerAdjoint.not_mem_of_not_mem_closure {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) {s : Set β} {P : β} (hP : Pl.toFun s) :
                            Ps
                            theorem LowerAdjoint.le_iff_subset {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (s : Set β) (S : α) :
                            l.toFun s S s S
                            theorem LowerAdjoint.mem_iff {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (s : Set β) (x : β) :
                            x l.toFun s ∀ (S : α), s Sx S
                            theorem LowerAdjoint.eq_of_le {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) {s : Set β} {S : α} (h₁ : s S) (h₂ : S l.toFun s) :
                            l.toFun s = S
                            theorem LowerAdjoint.closure_union_closure_subset {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                            (l.toFun x) (l.toFun y) (l.toFun (x y))
                            @[simp]
                            theorem LowerAdjoint.closure_union_closure_left {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                            l.toFun ((l.toFun x) y) = l.toFun (x y)
                            @[simp]
                            theorem LowerAdjoint.closure_union_closure_right {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                            l.toFun (x (l.toFun y)) = l.toFun (x y)
                            theorem LowerAdjoint.closure_union_closure {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                            l.toFun ((l.toFun x) (l.toFun y)) = l.toFun (x y)
                            @[simp]
                            theorem LowerAdjoint.closure_iUnion_closure {α : Type u_1} {ι : Sort u_2} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (f : ια) :
                            l.toFun (⋃ (i : ι), (l.toFun (f i))) = l.toFun (⋃ (i : ι), (f i))
                            @[simp]
                            theorem LowerAdjoint.closure_iUnion₂_closure {α : Type u_1} {ι : Sort u_2} {κ : ιSort u_3} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (f : (i : ι) → κ iα) :
                            l.toFun (⋃ (i : ι), ⋃ (j : κ i), (l.toFun (f i j))) = l.toFun (⋃ (i : ι), ⋃ (j : κ i), (f i j))

                            Translations between GaloisConnection, LowerAdjoint, ClosureOperator #

                            @[simp]
                            theorem GaloisConnection.lowerAdjoint_toFun {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) :
                            ∀ (a : α), (GaloisConnection.lowerAdjoint gc).toFun a = l a
                            def GaloisConnection.lowerAdjoint {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) :

                            Every Galois connection induces a lower adjoint.

                            Equations
                            Instances For
                              @[simp]
                              theorem GaloisConnection.closureOperator_isClosed {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) (x : α) :
                              (GaloisConnection.closureOperator gc).IsClosed x = (u (l x) = x)
                              @[simp]
                              theorem GaloisConnection.closureOperator_apply {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) (x : α) :
                              def GaloisConnection.closureOperator {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) :

                              Every Galois connection induces a closure operator given by the composition. This is the partial order version of the statement that every adjunction induces a monad.

                              Equations
                              Instances For

                                The set of closed elements has a Galois insertion to the underlying type.

                                Equations
                                Instances For
                                  @[simp]

                                  The Galois insertion associated to a closure operator can be used to reconstruct the closure operator. Note that the inverse in the opposite direction does not hold in general.