Documentation

Mathlib.Algebra.Category.BoolRing

The category of Boolean rings #

This file defines BoolRing, the category of Boolean rings.

TODO #

Finish the equivalence with BoolAlg.

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def BoolRing :
Type (u_1 + 1)

The category of Boolean rings.

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    instance BoolRing.instCoeSortBoolRingType :
    CoeSort BoolRing (Type u_1)
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    instance BoolRing.instBooleanRingα (X : BoolRing) :
    BooleanRing ↑X
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    def BoolRing.of (α : Type u_1) [BooleanRing α] :
    BoolRing

    Construct a bundled BoolRing from a BooleanRing.

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      @[simp]
      theorem BoolRing.coe_of (α : Type u_1) [BooleanRing α] :
      ↑(BoolRing.of α) = α
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      instance BoolRing.instInhabitedBoolRing :
      Inhabited BoolRing
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      instance BoolRing.instParentProjectionTypeCommRingBooleanRingToCommRing :
      CategoryTheory.BundledHom.ParentProjection @BooleanRing.toCommRing
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      instance BoolRing.instConcreteCategoryBoolRingInstBoolRingLargeCategory :
      CategoryTheory.ConcreteCategory BoolRing
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      instance BoolRing.hasForgetToCommRing :
      CategoryTheory.HasForget₂ BoolRing CommRingCat
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      • One or more equations did not get rendered due to their size.
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      @[simp]
      theorem BoolRing.Iso.mk_inv {α : BoolRing} {β : BoolRing} (e : ↑α ≃+* ↑β) :
      (BoolRing.Iso.mk e).inv = ↑(RingEquiv.symm e)
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      @[simp]
      theorem BoolRing.Iso.mk_hom {α : BoolRing} {β : BoolRing} (e : ↑α ≃+* ↑β) :
      (BoolRing.Iso.mk e).hom = ↑e
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      def BoolRing.Iso.mk {α : BoolRing} {β : BoolRing} (e : ↑α ≃+* ↑β) :
      α ≅ β

      Constructs an isomorphism of Boolean rings from a ring isomorphism between them.

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        Equivalence between BoolAlg and BoolRing #

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        @[simp]
        theorem BoolRing.hasForgetToBoolAlg_forget₂_map {X : BoolRing} {Y : BoolRing} (f : ↑X →+* ↑Y) :
        CategoryTheory.HasForget₂.forget₂.toPrefunctor.map f = RingHom.asBoolAlg f
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        @[simp]
        theorem BoolRing.hasForgetToBoolAlg_forget₂_obj (X : BoolRing) :
        CategoryTheory.HasForget₂.forget₂.toPrefunctor.obj X = BoolAlg.of (AsBoolAlg ↑X)
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        instance BoolRing.hasForgetToBoolAlg :
        CategoryTheory.HasForget₂ BoolRing BoolAlg
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        • One or more equations did not get rendered due to their size.
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        instance instBooleanAlgebraαLatticeToLatToBddLatToBddDistLat {X : BoolAlg} :
        BooleanAlgebra ↑(BddDistLat.toBddLat (BoolAlg.toBddDistLat X)).toLat
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        @[simp]
        theorem BoolAlg.hasForgetToBoolRing_forget₂_obj (X : BoolAlg) :
        CategoryTheory.HasForget₂.forget₂.toPrefunctor.obj X = BoolRing.of (AsBoolRing ↑X)
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        @[simp]
        theorem BoolAlg.hasForgetToBoolRing_forget₂_map {X : BoolAlg} {Y : BoolAlg} (f : BoundedLatticeHom ↑(BddDistLat.toBddLat (BoolAlg.toBddDistLat X)).toLat ↑(BddDistLat.toBddLat (BoolAlg.toBddDistLat Y)).toLat) :
        CategoryTheory.HasForget₂.forget₂.toPrefunctor.map f = BoundedLatticeHom.asBoolRing f
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        instance BoolAlg.hasForgetToBoolRing :
        CategoryTheory.HasForget₂ BoolAlg BoolRing
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        • One or more equations did not get rendered due to their size.
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        @[simp]
        theorem boolRingCatEquivBoolAlg_functor :
        boolRingCatEquivBoolAlg.functor = CategoryTheory.forget₂ BoolRing BoolAlg
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        @[simp]
        theorem boolRingCatEquivBoolAlg_inverse :
        boolRingCatEquivBoolAlg.inverse = CategoryTheory.forget₂ BoolAlg BoolRing
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        def boolRingCatEquivBoolAlg :
        BoolRing ≌ BoolAlg

        The equivalence between Boolean rings and Boolean algebras. This is actually an isomorphism.

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