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8 The (stable) proset/poset of a category
 8.1 Global variables
 8.2 Constructors
 8.3 Attributes
 8.4 Examples
 8.5 GAP categories

8 The (stable) proset/poset of a category

8.1 Global variables

8.1-1 CAP_INTERNAL_METHOD_NAME_LIST_FOR_PREORDERED_SET_OF_CATEGORY
‣ CAP_INTERNAL_METHOD_NAME_LIST_FOR_PREORDERED_SET_OF_CATEGORY( global variable )

8.2 Constructors

8.2-1 CreateProsetOrPosetOfCategory
‣ CreateProsetOrPosetOfCategory( C )( operation )

8.2-2 ProsetOfCategory
‣ ProsetOfCategory( C )( attribute )
gap> LoadPackage( "Locales", false );
true
gap> q := FinQuiver( "q(a,b)[f:a->b,g:b->a]" );
FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" )
gap> F := PathCategory( q );
PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
gap> C := F / [ [ F.fg, F.id_a ], [ F.gf, F.id_b ] ];
PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
/ [ f⋅g = id(a), g⋅f = id(b) ]
gap> P := ProsetOfCategory( C );
ProsetOfCategory( PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
/ [ f⋅g = id(a), g⋅f = id(b) ] )
gap> Display( P );
A CAP category with name
ProsetOfCategory( PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
/ [ f⋅g = id(a), g⋅f = id(b) ] ):

11 primitive operations were used to derive 76 operations for this category
which algorithmically
* IsThinCategory
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsFiniteCategory
* IsEquippedWithHomomorphismStructure
gap> SetOfObjects( P );
[ An object in the proset given by: (a), An object in the proset given by: (b) ]
gap> P.a = P.b;
false
gap> Length( SetOfMorphisms( P ) );
4
gap> Size( P );
4
gap> SetOfGeneratingMorphisms( P );
[ A morphism in the proset given by: [f]:(a) -≻ (b),
  A morphism in the proset given by: [g]:(b) -≻ (a) ]

8.2-3 StableProsetOfCategory
‣ StableProsetOfCategory( C )( attribute )

8.2-4 PosetOfCategory
‣ PosetOfCategory( C )( attribute )
gap> LoadPackage( "Locales", false );
true
gap> q := FinQuiver( "q(a,b)[f:a->b,g:b->a]" );
FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" )
gap> F := PathCategory( q );
PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
gap> C := F / [ [ F.fg, F.id_a ], [ F.gf, F.id_b ] ];
PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
/ [ f⋅g = id(a), g⋅f = id(b) ]
gap> P := PosetOfCategory( C );
PosetOfCategory( PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
/ [ f⋅g = id(a), g⋅f = id(b) ] )
gap> Display( P );
A CAP category with name
PosetOfCategory( PathCategory( FinQuiver( "q(a,b)[f:a-≻b,g:b-≻a]" ) )
/ [ f⋅g = id(a), g⋅f = id(b) ] ):

9 primitive operations were used to derive 78 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsFiniteCategory
* IsPosetCategory
* IsEquippedWithHomomorphismStructure
gap> SetOfObjects( P );
[ An object in the poset given by: (a) ]
gap> P.a = P.b;
true
gap> Length( SetOfMorphisms( P ) );
1
gap> Size( P );
1
gap> SetOfGeneratingMorphisms( P );
[  ]

8.2-5 StablePosetOfCategory
‣ StablePosetOfCategory( C )( attribute )

8.3 Attributes

8.3-1 AmbientCategory
‣ AmbientCategory( P )( attribute )

Returns: a list

The ambient category of the proset P.

8.3-2 UnderlyingObject
‣ UnderlyingObject( obj )( attribute )

Returns: a CAP category object

The object in the ambient category underlying obj.

8.3-3 UnderlyingMorphism
‣ UnderlyingMorphism( mor )( attribute )

Returns: a CAP category morphism

The morphism in the ambient category underlying mor.

8.4 Examples

8.4-1 Ideal quotient
gap> LoadPackage( "FreydCategoriesForCAP" );
true
gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.04.16" );
true
gap> LoadPackage( "LazyCategories", ">= 2020.05.01" );
true
gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := Q["x,y"];
Q[x,y]
gap> K := HomalgMatrix( "[ x, -x ]", 2, 1, R );
<A 2 x 1 matrix over an external ring>
gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R );
<A 2 x 1 matrix over an external ring>
gap> I := KroneckerMat( K, J );
<An unevaluated 4 x 1 matrix over an external ring>
gap> P := CategoryOfRows( R );
Rows( Q[x,y] )
gap> P := LazyCategory( P : show_evaluation := false );
LazyCategory( Rows( Q[x,y] ) )
gap> I := I / P; SetLabel( I, "I" ); SetLabel( Source( I ), "R^i" );
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> R := Target( I ); SetLabel( R, "R" );
<An evaluated object in LazyCategory( Rows( Q[x,y] ) )>
gap> J := J / P; SetLabel( J, "J" ); SetLabel( Source( J ), "R^j" );
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> K := K / P;
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> S := SliceCategoryOverTensorUnit( P );
SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) )
gap> L := PosetOfCategory( S );
PosetOfCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )
gap> I := I / S / L;
An object in the poset given by: An object in the slice category given by:
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> J := J / S / L;
An object in the poset given by: An object in the slice category given by:
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> K := K / S / L;
An object in the poset given by: An object in the slice category given by:
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> IqJ := InternalHom( J, I ); ## this is the ideal quotient I : J
An object in the poset given by: An object in the slice category given by:
<A morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> K = IqJ;
true
gap> LoadPackage( "FreydCategoriesForCAP" );
true
gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.04.16" );
true
gap> LoadPackage( "LazyCategories", ">= 2020.05.01" );
true
gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := Q["x,y"];
Q[x,y]
gap> K := HomalgMatrix( "[ x, -x ]", 2, 1, R );
<A 2 x 1 matrix over an external ring>
gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R );
<A 2 x 1 matrix over an external ring>
gap> I := KroneckerMat( K, J );
<An unevaluated 4 x 1 matrix over an external ring>
gap> P := CategoryOfRows( R );
Rows( Q[x,y] )
gap> P := LazyCategory( P : primitive_operations := true, show_evaluation := false );
LazyCategory( Rows( Q[x,y] ) )
gap> I := I / P; SetLabel( I, "I" ); SetLabel( Source( I ), "R^i" );
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> R := Target( I ); SetLabel( R, "R" );
<An evaluated object in LazyCategory( Rows( Q[x,y] ) )>
gap> J := J / P; SetLabel( J, "J" ); SetLabel( Source( J ), "R^j" );
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> K := K / P;
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> S := SliceCategoryOverTensorUnit( P );
SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) )
gap> L := PosetOfCategory( S );
PosetOfCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )
gap> I := I / S / L;
An object in the poset given by: An object in the slice category given by:
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> J := J / S / L;
An object in the poset given by: An object in the slice category given by:
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> K := K / S / L;
An object in the poset given by: An object in the slice category given by:
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> IqJ := InternalHom( J, I ); ## this is the ideal quotient I : J
An object in the poset given by: An object in the slice category given by:
<A morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> K = IqJ;
true
gap> LoadPackage( "FreydCategoriesForCAP" );
true
gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.04.16" );
true
gap> LoadPackage( "LazyCategories", ">= 2020.06.01" );
true
gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := Q["x,y"];
Q[x,y]
gap> K := HomalgMatrix( "[ x, -x ]", 2, 1, R );
<A 2 x 1 matrix over an external ring>
gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R );
<A 2 x 1 matrix over an external ring>
gap> I := KroneckerMat( K, J );
<An unevaluated 4 x 1 matrix over an external ring>
gap> P := CategoryOfRows( R );
Rows( Q[x,y] )
gap> P := LazyCategory( P : show_evaluation := false );
LazyCategory( Rows( Q[x,y] ) )
gap> I := I / P; SetLabel( I, "I" ); SetLabel( Source( I ), "R^i" );
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> R := Target( I ); SetLabel( R, "R" );
<An evaluated object in LazyCategory( Rows( Q[x,y] ) )>
gap> J := J / P; SetLabel( J, "J" ); SetLabel( Source( J ), "R^j" );
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> K := K / P;
<An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )>
gap> S := LazyCategory( SliceCategoryOverTensorUnit( P ) );
LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )
gap> L := LazyCategory( PosetOfCategory( S ) );
LazyCategory( PosetOfCategory( LazyCategory(
SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )
gap> I := I / S; SetLabel( I, "I" );
<An evaluated object in LazyCategory(
 SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )>
gap> I := I / L; SetLabel( I, "I" );
<An evaluated object in LazyCategory( PosetOfCategory( LazyCategory(
 SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )>
gap> J := J / S; SetLabel( J, "J" );
<An evaluated object in LazyCategory(
 SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )>
gap> J := J / L; SetLabel( J, "J" );
<An evaluated object in LazyCategory( PosetOfCategory( LazyCategory(
 SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )>
gap> K := K / S; SetLabel( K, "K" );
<An evaluated object in LazyCategory(
 SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )>
gap> K := K / L; SetLabel( K, "K" );
<An evaluated object in LazyCategory( PosetOfCategory( LazyCategory(
 SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )>
gap> IqJ := InternalHom( J, I ); ## this is the ideal quotient I : J
<An object in LazyCategory( PosetOfCategory( LazyCategory(
 SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )>
gap> IsHomSetInhabited( K, IqJ );
true
gap> IsHomSetInhabited( IqJ, K );
true

8.5 GAP categories

8.5-1 Poset of category
gap> LoadPackage( "SubcategoriesForCAP", ">= 2024.02-11", false );
true
gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := Q["x,y"];
Q[x,y]
gap> F := CategoryOfRows( R );
Rows( Q[x,y] )
gap> S := SliceCategoryOverTensorUnit( F );
SliceCategoryOverTensorUnit( Rows( Q[x,y] ) )
gap> P := PosetOfCategory( S );
PosetOfCategory( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) )
gap> Display( P );
A CAP category with name
PosetOfCategory( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ):

17 primitive operations were used to derive 267 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsEquippedWithHomomorphismStructure
* IsSymmetricClosedMonoidalLattice
gap> I := HomalgMatrix( "[ x ]", 1, 1, R ) / F / S / P;
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R ) / F / S / P;
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IJ := TensorProduct( I, J );
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IiJ := DirectProduct( I, J );
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IiJ = I;
true
gap> IsHomSetInhabited( IJ, IiJ );
true
gap> IsHomSetInhabited( IiJ, IJ );
false
gap> IpJ := Coproduct( I, J );
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IpJ = J;
true
gap> IJqJ := InternalHom( J, IJ ); ## this is the ideal quotient IJ : J
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IJqJ = I;
true
gap> iota := InternalHom( UniversalMorphismIntoTerminalObject( J ), IJ );
<An epi-, monomorphism in PosetOfCategory( SliceCategoryOverTensorUnit( \
 Rows( Q[x,y] ) ) )>
gap> IsWellDefined( iota );
true
gap> IsIsomorphism( iota );
false
gap> IJJ := TensorProduct( IJ, J );
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IJJqJ := InternalHom( J, IJJ );
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IJJqJ = I;
false
gap> IJJsJ := StableInternalHom( J, IJJ );
An object in the poset given by: An object in the slice category given by:
<A morphism in Rows( Q[x,y] )>
gap> IJJsJ = I;
true

8.5-2 Stable poset of category
gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.10-02" );
true
gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := Q["x"] / "x^2";
Q[x]/( x^2 )
gap> F := CategoryOfRows( R );
Rows( Q[x]/( x^2 ) )
gap> S := SliceCategoryOverTensorUnit( F );
SliceCategoryOverTensorUnit( Rows( Q[x]/( x^2 ) ) )
gap> P := PosetOfCategory( S );
PosetOfCategory( SliceCategoryOverTensorUnit( Rows( Q[x]/( x^2 ) ) ) )
gap> L := StablePosetOfCategory( P );
StablePosetOfCategory( PosetOfCategory(
SliceCategoryOverTensorUnit( Rows( Q[x]/( x^2 ) ) ) ) )
gap> N := HomalgMatrix( "[ x ]", 1, 1, R ) / F / S / P;
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x]/( x^2 ) )>
gap> I := InitialObject( P );
An object in the poset given by:
An object in the slice category given by:
<A zero, split monomorphism in Rows( Q[x]/( x^2 ) )>
gap> I = N;
false
gap> IsInitial( I );
true
gap> IsInitial( N );
false
gap> rN := N / L;
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x]/( x^2 ) )>
gap> rI := I / L;
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by:
<A zero, split monomorphism in Rows( Q[x]/( x^2 ) )>
gap> rI = rN;
true
gap> IsInitial( rI );
true
gap> IsInitial( rN );
true
gap> LoadPackage( "SubcategoriesForCAP", ">= 2024.02-11", false );
true
gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := Q["x,y"];
Q[x,y]
gap> F := CategoryOfRows( R );
Rows( Q[x,y] )
gap> S := SliceCategoryOverTensorUnit( F );
SliceCategoryOverTensorUnit( Rows( Q[x,y] ) )
gap> P := PosetOfCategory( S );
PosetOfCategory( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) )
gap> Display( P );
A CAP category with name
PosetOfCategory( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ):

17 primitive operations were used to derive 267 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsEquippedWithHomomorphismStructure
* IsSymmetricClosedMonoidalLattice
gap> L := StablePosetOfCategory( P );
StablePosetOfCategory( PosetOfCategory(
SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ) )
gap> Display( L );
A CAP category with name
StablePosetOfCategory( PosetOfCategory(
SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ) ):

16 primitive operations were used to derive 329 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsEquippedWithHomomorphismStructure
* IsHeytingAlgebra
* IsSymmetricClosedMonoidalLattice
and furthermore mathematically
* IsStableProset
gap> I := HomalgMatrix( "[ x ]", 1, 1, R ) / F / S / P / L;
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R ) / F / S / P / L;
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IJ := TensorProduct( I, J );
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IiJ := DirectProduct( I, J );
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IsHomSetInhabited( IJ, IiJ );
true
gap> IsHomSetInhabited( IiJ, IJ );
true
gap> IiJ = I;
true
gap> IpJ := Coproduct( I, J );
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IpJ = J;
true
gap> IJqJ := InternalHom( J, IJ ); ## this is the ideal quotient IJ : J
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IJqJ = I;
true
gap> iota := InternalHom( UniversalMorphismIntoTerminalObject( J ), IJ );
A morphism in the stable poset given by: <An epi-, monomorphism in
PosetOfCategory( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) )>
gap> IsWellDefined( iota );
true
gap> IsOne( iota );
true
gap> IJJ := TensorProduct( IJ, J );
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IJJqJ := InternalHom( J, IJJ );
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IJJqJ = I;
true
gap> IJJsJ := StableInternalHom( J, IJJ );
An object in the stable poset given by:
An object in the poset given by:
An object in the slice category given by: <A morphism in Rows( Q[x,y] )>
gap> IJJsJ = I;
true

8.5-3 IsProsetOrPosetOfCapCategory
‣ IsProsetOrPosetOfCapCategory( arg )( filter )

Returns: true or false

The GAP category of a proset or poset of a CAP category.

8.5-4 IsCellInProsetOrPosetOfCategory
‣ IsCellInProsetOrPosetOfCategory( arg )( filter )

Returns: true or false

The GAP category of cells in a proset or poset of a CAP category.

8.5-5 IsCellInStableProsetOrPosetOfCategory
‣ IsCellInStableProsetOrPosetOfCategory( arg )( filter )

Returns: true or false

The GAP category of cells in a stable proset or poset of a CAP category.

8.5-6 IsObjectInProsetOrPosetOfCategory
‣ IsObjectInProsetOrPosetOfCategory( arg )( filter )

Returns: true or false

The GAP category of objects in a proset or poset of a CAP category.

8.5-7 IsProsetOfCapCategory
‣ IsProsetOfCapCategory( arg )( filter )

Returns: true or false

The GAP category of a proset of a CAP category.

8.5-8 IsCellInProsetOfCategory
‣ IsCellInProsetOfCategory( arg )( filter )

Returns: true or false

The GAP category of cells in a proset of a CAP category.

8.5-9 IsObjectInProsetOfCategory
‣ IsObjectInProsetOfCategory( arg )( filter )

Returns: true or false

The GAP category of objects in a proset of a CAP category.

8.5-10 IsMorphismInProsetOfCategory
‣ IsMorphismInProsetOfCategory( arg )( filter )

Returns: true or false

The GAP category of morphisms in a proset of a CAP category.

8.5-11 IsPosetOfCapCategory
‣ IsPosetOfCapCategory( arg )( filter )

Returns: true or false

The GAP category of a poset of a CAP category.

8.5-12 IsCellInPosetOfCategory
‣ IsCellInPosetOfCategory( arg )( filter )

Returns: true or false

The GAP category of cells in a poset of a CAP category.

8.5-13 IsObjectInPosetOfCategory
‣ IsObjectInPosetOfCategory( arg )( filter )

Returns: true or false

The GAP category of objects in a poset of a CAP category.

8.5-14 IsMorphismInPosetOfCategory
‣ IsMorphismInPosetOfCategory( arg )( filter )

Returns: true or false

The GAP category of morphisms in a poset of a CAP category.

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