‣ IsCategoryOfFiniteSets( object ) | ( filter ) |
Returns: true or false
The GAP category of categories of finite sets.
‣ IsObjectInCategoryOfFiniteSets( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the category of finite sets.
‣ IsMorphismInCategoryOfFiniteSets( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the category of finite sets.
‣ AsList( M ) | ( attribute ) |
Returns: a GAP set
The GAP set of the list used to construct a finite set \(S\), i.e., AsList( FinSet( L ) ) = Set( L ).
‣ Length( M ) | ( attribute ) |
Returns: an integer
The length of the GAP set of the list used to construct a finite set \(S\), i.e., Length( FinSet( L ) ) = Length( Set( L ) ).
‣ AsList( f ) | ( attribute ) |
Returns: a list
The relation underlying a map between finite sets, i.e., AsList( MapOfFinSets( S, G, T ) ) = G.
‣ CategoryOfFiniteSets( ) | ( operation ) |
Returns: a CAP category
Construct a category of finite sets.
‣ FinSets | ( global variable ) |
The default instance of the category of finite sets. It is automatically created while loading this package.
‣ FinSet( cat_of_fin_sets, L ) | ( operation ) |
Returns: a CAP object
Construct a finite set out of the dense list L, i.e., an object in the CAP category cat_of_fin_sets. The GAP operation Set must be applicable to L without throwing an error. Equality is determined as follows: FinSet( L1 ) = FinSet( L2 ) iff IsEqualForElementsOfFinSets( Immutable( Set( L1 ) ), Immutable( Set( L2 ) ) ). Warning: all internal operations use FinSetNC (see below) instead of FinSet. Thus, this notion of equality is only valid for objects created by calling FinSet explicitly. Internally, FinSet( cat_of_fin_sets, L ) is an alias for FinSetNC( cat_of_fin_sets, Set( L ) ) and equality is determined as for FinSetNC. Thus, FinSet( cat_of_fin_sets, L1 ) = FinSetNC( cat_of_fin_sets, L2 ) iff IsEqualForElementsOfFinSets( Immutable( Set( L1 ) ), Immutable( L2 ) ) and FinSetNC( cat_of_fin_sets, L1 ) = FinSet( cat_of_fin_sets, L2 ) iff IsEqualForElementsOfFinSets( Immutable( L1 ), Immutable( Set( L2 ) ) ).
‣ FinSet( L ) | ( operation ) |
Returns: a CAP object
Return FinSet( FinSets, L ).
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 3, 2, 2, 1 ] ); <An object in FinSets> gap> Display( S ); [ 1, 2, 3 ] gap> L := AsList( S );; gap> Display( L ); [ 1, 2, 3 ] gap> Q := FinSet( L ); <An object in FinSets> gap> S = Q; true gap> FinSet( [ 1, 2 ] ) = FinSet( [ 2, 1 ] ); true gap> M := FinSetNC( [ 1, 2, 3, 3 ] ); <An object in FinSets> gap> IsWellDefined( M ); false
‣ FinSetNC( cat_of_fin_sets, L ) | ( operation ) |
Returns: a CAP object
Construct a finite set out of the duplicate-free (w.r.t. IsEqualForElementsOfFinSets) and dense list L, i.e., an object in the CAP category cat_of_fin_sets,. Equality is determined as follows: FinSetNC( cat_of_fin_sets, L1 ) = FinSetNC( cat_of_fin_sets, L2 ) iff IsEqualForElementsOfFinSets( Immutable( L1 ), Immutable( L2 ) ).
‣ FinSetNC( L ) | ( operation ) |
Returns: a CAP object
Return FinSetNC( FinSets, L ).
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSetNC( [ 1, 3, 2 ] ); <An object in FinSets> gap> Display( S ); [ 1, 3, 2 ] gap> L := AsList( S );; gap> Display( L ); [ 1, 3, 2 ] gap> Q := FinSetNC( L ); <An object in FinSets> gap> S = Q; true gap> S := ObjectConstructor( FinSets, [ 1, 3, 2 ] ); <An object in FinSets> gap> Display( S ); [ 1, 3, 2 ] gap> L := ObjectDatum( S );; gap> Display( L ); [ 1, 3, 2 ] gap> Q := ObjectConstructor( FinSets, L ); <An object in FinSets> gap> S = Q; true gap> FinSetNC( [ 1, 2 ] ) = FinSetNC( [ 2, 1 ] ); false
‣ MapOfFinSets( S, G, T ) | ( operation ) |
Returns: a CAP morphism
Construct a map \(\phi:\)S\(\to\)T of the finite sets S and T, i.e., a morphism in the CAP category FinSets, where G is a dense list of pairs in S\(\times\)T describing the graph of \(\phi\).
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 3, 2, 2, 1 ] ); <An object in FinSets> gap> T := FinSet( [ "a", "b", "c" ] ); <An object in FinSets> gap> G := [ [ 1, "b" ], [ 3, "b" ], [ 2, "a" ] ];; gap> phi := MapOfFinSets( S, G, T ); <A morphism in FinSets> gap> IsWellDefined( phi ); true gap> Display( AsList( phi ) ); [ [ 1, "b" ], [ 2, "a" ], [ 3, "b" ] ] gap> phi2 := MorphismConstructor( S, G, T ); <A morphism in FinSets> gap> Display( MorphismDatum( phi2 ) ); [ [ 1, "b" ], [ 2, "a" ], [ 3, "b" ] ] gap> phi = phi2; true gap> phi( 1 ); "b" gap> phi( 2 ); "a" gap> phi( 3 ); "b" gap> Display( List( S, phi ) ); [ "b", "a", "b" ] gap> psi := [ [ 1, "b" ], [ 2, "a" ], [ 3, "b" ] ];; gap> psi := MapOfFinSets( S, psi, T ); <A morphism in FinSets> gap> IsWellDefined( psi ); true gap> phi = psi; true gap> psi := MapOfFinSets( S, [ [ 1, "d" ], [ 3, "b" ] ], T ); <A morphism in FinSets> gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( S, [ 1, 2, 3 ], T ); <A morphism in FinSets> gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( S, [ [ 1, "b" ], [ 3, "b" ], [ 2, "a", "b" ] ], T ); <A morphism in FinSets> gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( S, [ [ 5, "b" ], [ 3, "b" ], [ 2, "a" ] ], T ); <A morphism in FinSets> gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( S, [ [ 1, "d" ], [ 3, "b" ], [ 2, "a" ] ], T ); <A morphism in FinSets> gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( S, [ [ 1, "b" ], [ 2, "b" ], [ 2, "a" ] ], T ); <A morphism in FinSets> gap> IsWellDefined( psi ); false
‣ MapOfFinSetsNC( S, G, T ) | ( operation ) |
Returns: a CAP morphism
Construct a map \(\phi:\)S\(\to\)T of the finite sets S and T, i.e., a morphism in the CAP category FinSets, where G is a duplicate-free and dense list of pairs in S\(\times\)T describing the graph of \(\phi\).
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSetNC( [ 1, 3, 2 ] ); <An object in FinSets> gap> T := FinSetNC( [ "a", "b", "c" ] ); <An object in FinSets> gap> G := [ [ 1, "b" ], [ 3, "b" ], [ 2, "a" ] ];; gap> phi := MapOfFinSetsNC( S, G, T ); <A morphism in FinSets> gap> IsWellDefined( phi ); true gap> phi( 1 ); "b" gap> phi( 2 ); "a" gap> phi( 3 ); "b" gap> Display( List( S, phi ) ); [ "b", "b", "a" ] gap> psi := [ [ 1, "b" ], [ 2, "a" ], [ 3, "b" ] ];; gap> psi := MapOfFinSetsNC( S, psi, T ); <A morphism in FinSets> gap> IsWellDefined( psi ); true gap> phi = psi; true
‣ IsEqualForElementsOfFinSets( a, b ) | ( operation ) |
Returns: a boolean
Compares two arbitrary objects using the following rules:
integers, strings and chars are compared using the operation =
dense lists and records are compared recursively
CAP category objects are compared using IsEqualForObjects (if available)
CAP category morphisms are compared using IsEqualForMorphismsOnMor (if available)
other objects are compared using IsIdenticalObj
Note: if CAP category objects or CAP category morphisms are compared using IsEqualForObjects or IsEqualForMorphismsOnMor, respectively, the result must not be fail.
gap> LoadPackage( "FinSetsForCAP", false ); true gap> IsEqualForElementsOfFinSets( 2, 2 ); true gap> IsEqualForElementsOfFinSets( 2, "2" ); false gap> IsEqualForElementsOfFinSets( 'a', 'a' ); true gap> IsEqualForElementsOfFinSets( 'a', 'b' ); false gap> IsEqualForElementsOfFinSets( [ 2 ], [ 2 ] ); true gap> IsEqualForElementsOfFinSets( [ 2 ], [ 2, 3 ] ); false gap> IsEqualForElementsOfFinSets( rec( a := "a", b := "b" ), > rec( b := "b", a := "a" ) > ); true gap> IsEqualForElementsOfFinSets( rec( a := "a", b := "b" ), > rec( a := "a" ) > ); false gap> IsEqualForElementsOfFinSets( rec( a := "a", b := "b" ), > rec( a := "a", b := "notb" ) > ); false gap> M := FinSet( [ ] );; gap> N := FinSet( [ ] );; gap> m := FinSet( 0 );; gap> id_M := IdentityMorphism( M );; gap> id_N := IdentityMorphism( N );; gap> id_m := IdentityMorphism( m );; gap> IsEqualForElementsOfFinSets( M, N ); true gap> IsEqualForElementsOfFinSets( M, m ); false gap> IsEqualForElementsOfFinSets( id_M, id_N ); true gap> IsEqualForElementsOfFinSets( id_M, id_m ); false gap> IsEqualForElementsOfFinSets( FinSets, SkeletalFinSets ); false
1.4-2 \in‣ \in( obj, M ) | ( operation ) |
Returns: a boolean
Returns true if there exists an element in AsList( M ) which is equal to obj w.r.t. IsEqualForElementsOfFinSets and false if not.
‣ []( M, i ) | ( operation ) |
Returns: an object
Returns the i-th entry of the GAP set of the list used to construct a finite set \(S\), i.e., FinSet( L )[ i ] = Set( L )[ i ].
‣ Iterator( M ) | ( operation ) |
Returns: an iterator
An iterator of the GAP set of the list used to construct a finite set \(S\), i.e., Iterator( FinSet( L ) ) = Iterator( Set( L ) ).
‣ UnionOfFinSets( cat_of_fin_sets, L ) | ( operation ) |
Returns: a CAP object
Compute the set-theoretic union of the elements of L, where L is a dense list of finite sets in the category cat_of_fin_sets.
‣ ListOp( M, f ) | ( operation ) |
Returns: a list
Returns List( AsList( M ), f ).
‣ FilteredOp( M, f ) | ( operation ) |
Returns: a list
Returns FinSetNC( Filtered( AsList( M ), f ) ).
‣ First( M, f ) | ( operation ) |
Returns: a list
Returns First( AsList( M ), f ).
‣ EmbeddingOfFinSets( S, T ) | ( operation ) |
Returns: a CAP morphism
Construct the embedding \(\iota:\)S\(\to\)T of the finite sets S and T, where S must be subset of T.
‣ ProjectionOfFinSets( S, T ) | ( operation ) |
Returns: a CAP morphism
Construct the projection \(\pi:\)S\(\to\)T of the finite sets S and T, where T is a partition of S.
‣ Preimage( f, T_ ) | ( operation ) |
Returns: a CAP object
Compute the preimage of T_ under the morphism f.
‣ ImageObject( f, S_ ) | ( operation ) |
Returns: a CAP object
Compute the image of S_ under the morphism f.
‣ CallFuncList( phi, L ) | ( operation ) |
Returns: a list
Returns the image of L[1] under the map phi assuming L[1] is an element of AsList( Source( phi ) ).
‣ ListOp( F, phi ) | ( operation ) |
Returns: a list
Returns List( AsList( F ), phi ).
gap> LoadPackage( "FinSetsForCAP", false ); true gap> L := FinSet( [ ] ); <An object in FinSets> gap> M := FinSet( [ 2 ] ); <An object in FinSets> gap> N := FinSet( [ 3 ] ); <An object in FinSets> gap> IsHomSetInhabited( L, L ); true gap> IsHomSetInhabited( M, L ); false gap> IsHomSetInhabited( L, M ); true gap> IsHomSetInhabited( M, N ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> L := FinSet( 0 ); |0| gap> M := FinSet( 2 ); |2| gap> N := FinSet( 3 ); |3| gap> Display( MorphismsOfExternalHom( L, L ) ); [ ∅ ⱶ[ ]→ ∅ ] gap> Display( MorphismsOfExternalHom( M, L ) ); [ ] gap> Display( MorphismsOfExternalHom( L, M ) ); [ ∅ ⱶ[ ]→ { 0, 1 } ] gap> Display( MorphismsOfExternalHom( M, N ) ); [ { 0, 1 } ⱶ[ 0, 0 ]→ { 0, 1, 2 }, { 0, 1 } ⱶ[ 1, 0 ]→ { 0, 1, 2 },\ { 0, 1 } ⱶ[ 2, 0 ]→ { 0, 1, 2 }, { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1, 2 },\ { 0, 1 } ⱶ[ 1, 1 ]→ { 0, 1, 2 }, { 0, 1 } ⱶ[ 2, 1 ]→ { 0, 1, 2 },\ { 0, 1 } ⱶ[ 0, 2 ]→ { 0, 1, 2 }, { 0, 1 } ⱶ[ 1, 2 ]→ { 0, 1, 2 },\ { 0, 1 } ⱶ[ 2, 2 ]→ { 0, 1, 2 } ]
gap> LoadPackage( "FinSetsForCAP", false ); true gap> L := FinSet( 0 ); |0| gap> M := FinSet( 2 ); |2| gap> Display( ExactCoverWithGlobalElements( L ) ); [ ] gap> g := ExactCoverWithGlobalElements( M );; gap> Display( g ); [ { 0 } ⱶ[ 0 ]→ { 0, 1 }, { 0 } ⱶ[ 1 ]→ { 0, 1 } ] gap> IsOne( UniversalMorphismFromCoproduct( g ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> T := FinSet( [ "a", "b" ] ); <An object in FinSets> gap> phi := [ [ 1, "b" ], [ 2, "a" ], [ 3, "b" ] ];; gap> phi := MapOfFinSets( S, phi, T ); <A morphism in FinSets> gap> psi := [ [ "a", 3 ], [ "b", 1 ] ];; gap> psi := MapOfFinSets( T, psi, S ); <A morphism in FinSets> gap> alpha := PreCompose( phi, psi ); <A morphism in FinSets> gap> Display( List( S, alpha ) ); [ 1, 3, 1 ] gap> IsOne( alpha ); false
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> T := S; <An object in FinSets> gap> phi := [ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ];; gap> phi := MapOfFinSets( S, phi, T ); <A morphism in FinSets> gap> I := ImageObject( phi ); <An object in FinSets> gap> Length( I ); 3 gap> IsMonomorphism( phi ); true gap> IsSplitMonomorphism( phi ); true gap> IsEpimorphism( phi ); true gap> IsSplitEpimorphism( phi ); true gap> iota := ImageEmbedding( phi ); <A monomorphism in FinSets> gap> pi := CoastrictionToImage( phi ); <An epimorphism in FinSets> gap> PreCompose( pi, iota ) = phi; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> M := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> IsInitial( M ); false gap> IsTerminal( M ); false gap> I := InitialObject( FinSets ); <An object in FinSets> gap> IsInitial( I ); true gap> IsTerminal( I ); false gap> iota := UniversalMorphismFromInitialObject( M ); <A morphism in FinSets> gap> Display( I ); [ ] gap> T := TerminalObject( FinSets ); <An object in FinSets> gap> Display( T ); [ [ ] ] gap> IsInitial( T ); false gap> IsTerminal( T ); true gap> pi := UniversalMorphismIntoTerminalObject( M ); <A morphism in FinSets> gap> IsIdenticalObj( Range( pi ), T ); true gap> t := FinSet( [ "Julia" ] ); <An object in FinSets> gap> pi_t := UniversalMorphismIntoTerminalObjectWithGivenTerminalObject( M, t ); <A morphism in FinSets> gap> Display( List( M, pi_t ) ); [ "Julia", "Julia", "Julia" ]
gap> LoadPackage( "FinSetsForCAP", false ); true gap> I := FinSet( [ ] ); <An object in FinSets> gap> T := FinSet( [ 1 ] ); <An object in FinSets> gap> M := FinSet( [ 2 ] ); <An object in FinSets> gap> IsProjective( I ); true gap> IsProjective( T ); true gap> IsProjective( M ); true gap> IsOne( EpimorphismFromSomeProjectiveObject( I ) ); true gap> IsOne( EpimorphismFromSomeProjectiveObject( M ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> I := FinSet( [ ] ); <An object in FinSets> gap> T := FinSet( [ 1 ] ); <An object in FinSets> gap> M := FinSet( [ 2 ] ); <An object in FinSets> gap> IsInjective( I ); false gap> IsInjective( T ); true gap> IsInjective( M ); true gap> IsIsomorphism( MonomorphismIntoSomeInjectiveObject( I ) ); false gap> IsMonomorphism( MonomorphismIntoSomeInjectiveObject( I ) ); true gap> IsOne( MonomorphismIntoSomeInjectiveObject( M ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> Length( S ); 3 gap> T := FinSet( [ "a", "b" ] ); <An object in FinSets> gap> Length( T ); 2 gap> P := DirectProduct( S, T ); <An object in FinSets> gap> Length( P ); 6 gap> Display( P ); [ [ 1, "a" ], [ 2, "a" ], [ 3, "a" ], [ 1, "b" ], [ 2, "b" ], [ 3, "b" ] ]
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> Length( S ); 3 gap> T := FinSet( [ "a", "b" ] ); <An object in FinSets> gap> Length( T ); 2 gap> C := Coproduct( T, S ); <An object in FinSets> gap> Length( C ); 5 gap> Display( C ); [ [ 1, "a" ], [ 1, "b" ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ] ] gap> M := FinSet( [ 1, 2, 3, 4, 5, 6, 7 ] ); <An object in FinSets> gap> N := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> P := FinSet( [ 1, 2, 3, 4 ] ); <An object in FinSets> gap> C := Coproduct( M, N, P ); <An object in FinSets> gap> Display( AsList( C ) ); [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ],\ [ 1, 7 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ],\ [ 3, 3 ], [ 3, 4 ] ] gap> iota1 := InjectionOfCofactorOfCoproduct( [ M, N, P ], 1 ); <A morphism in FinSets> gap> IsWellDefined( iota1 ); true gap> Display( AsList( iota1 ) ); [ [ 1, [ 1, 1 ] ], [ 2, [ 1, 2 ] ], [ 3, [ 1, 3 ] ], [ 4, [ 1, 4 ] ],\ [ 5, [ 1, 5 ] ], [ 6, [ 1, 6 ] ], [ 7, [ 1, 7 ] ] ] gap> iota2 := InjectionOfCofactorOfCoproduct( [ M, N, P ], 2 ); <A morphism in FinSets> gap> IsWellDefined( iota2 ); true gap> Display( AsList( iota2 ) ); [ [ 1, [ 2, 1 ] ], [ 2, [ 2, 2 ] ], [ 3, [ 2, 3 ] ] ] gap> iota3 := InjectionOfCofactorOfCoproduct( [ M, N, P ], 3 ); <A morphism in FinSets> gap> IsWellDefined( iota3 ); true gap> Display( AsList( iota3 ) ); [ [ 1, [ 3, 1 ] ], [ 2, [ 3, 2 ] ], [ 3, [ 3, 3 ] ], [ 4, [ 3, 4 ] ] ] gap> psi := UniversalMorphismFromCoproduct( [ M, N, P ], > [ iota1, iota2, iota3 ] > ); <A morphism in FinSets> gap> psi = IdentityMorphism( Coproduct( [ M, N, P ] ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> T := FinSet( [ "a", "b", "c" ] ); <An object in FinSets> gap> phi := [ [ 1, "b" ], [ 2, "a" ], [ 3, "b" ] ];; gap> phi := MapOfFinSets( S, phi, T ); <A morphism in FinSets> gap> I := ImageObject( phi ); <An object in FinSets> gap> Length( I ); 2 gap> IsMonomorphism( phi ); false gap> IsSplitMonomorphism( phi ); false gap> IsEpimorphism( phi ); false gap> IsSplitEpimorphism( phi ); false gap> iota := ImageEmbedding( phi ); <A monomorphism in FinSets> gap> pi := CoastrictionToImage( phi ); <An epimorphism in FinSets> gap> PreCompose( pi, iota ) = phi; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> T := FinSet( [ "a", "b", "c" ] ); <An object in FinSets> gap> phi := [ [ 1, "b" ], [ 2, "a" ], [ 3, "b" ] ];; gap> phi := MapOfFinSets( S, phi, T ); <A morphism in FinSets> gap> I := CoimageObject( phi ); <An object in FinSets> gap> Length( I ); 2 gap> IsMonomorphism( phi ); false gap> IsSplitMonomorphism( phi ); false gap> IsEpimorphism( phi ); false gap> IsSplitEpimorphism( phi ); false gap> pi := CoimageProjection( phi ); <An epimorphism in FinSets> gap> iota := AstrictionToCoimage( phi ); <A morphism in FinSets> gap> PreCompose( pi, iota ) = phi; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1 .. 5 ] ); <An object in FinSets> gap> T := FinSet( [ 1 .. 3 ] ); <An object in FinSets> gap> f1 := MapOfFinSets( S, [ [1,3],[2,3],[3,1],[4,2],[5,2] ], T ); <A morphism in FinSets> gap> f2 := MapOfFinSets( S, [ [1,3],[2,2],[3,3],[4,1],[5,2] ], T ); <A morphism in FinSets> gap> f3 := MapOfFinSets( S, [ [1,3],[2,1],[3,2],[4,1],[5,2] ], T ); <A morphism in FinSets> gap> D := [ f1, f2, f3 ];; gap> Eq := Equalizer( D ); <An object in FinSets> gap> Display( Eq ); [ 1, 5 ] gap> iota := EmbeddingOfEqualizer( D );; gap> IsWellDefined( iota ); true gap> Im := ImageObject( iota ); <An object in FinSets> gap> Display( Im ); [ 1, 5 ] gap> mu := MorphismFromEqualizerToSink( D );; gap> PreCompose( iota, f1 ) = mu; true gap> M := FinSet( [ "a" ] );; gap> phi := MapOfFinSets( M, [ [ "a", 5 ] ], S );; gap> IsWellDefined( phi ); true gap> psi := UniversalMorphismIntoEqualizer( D, phi ); <A morphism in FinSets> gap> IsWellDefined( psi ); true gap> Display( psi ); [ [ "a" ], [ [ "a", 5 ] ], [ 1, 5 ] ] gap> PreCompose( psi, iota ) = phi; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> M := FinSet( [ 1 .. 5 ] ); <An object in FinSets> gap> N1 := FinSet( [ 1 .. 3 ] ); <An object in FinSets> gap> iota1 := EmbeddingOfFinSets( N1, M ); <A monomorphism in FinSets> gap> N2 := FinSet( [ 2 .. 5 ] ); <An object in FinSets> gap> iota2 := EmbeddingOfFinSets( N2, M ); <A monomorphism in FinSets> gap> D := [ iota1, iota2 ];; gap> int := FiberProduct( D ); <An object in FinSets> gap> Display( int ); [ [ 2, 2 ], [ 3, 3 ] ] gap> pi1 := ProjectionInFactorOfFiberProduct( D, 1 ); <A monomorphism in FinSets> gap> int1 := ImageObject( pi1 ); <An object in FinSets> gap> Display( int1 ); [ 2, 3 ] gap> pi2 := ProjectionInFactorOfFiberProduct( D, 2 ); <A monomorphism in FinSets> gap> int2 := ImageObject( pi2 ); <An object in FinSets> gap> Display( int2 ); [ 2, 3 ]
gap> LoadPackage( "FinSetsForCAP", false ); true gap> N := FinSet( [1,3] ); <An object in FinSets> gap> M := FinSet( [1,2,4] ); <An object in FinSets> gap> f := MapOfFinSets( N, [ [1,1], [3,2] ], M ); <A morphism in FinSets> gap> g := MapOfFinSets( N, [ [1,2], [3,4] ], M ); <A morphism in FinSets> gap> C := Coequalizer( f, g ); <An object in FinSets> gap> Display( AsList( C ) ); [ [ 1, 2, 4 ] ] gap> A := FinSet( [ 1, 2, 3, 4 ] ); <An object in FinSets> gap> B := FinSet( [ 1, 2, 3, 4, 5, 6, 7, 8 ] ); <An object in FinSets> gap> f1 := MapOfFinSets( A, [ [ 1, 1 ], [ 2, 2 ], [ 3, 3 ], [ 4, 8 ] ], B ); <A morphism in FinSets> gap> f2 := MapOfFinSets( A, [ [ 1, 2 ], [ 2, 3 ], [ 3, 8 ], [ 4, 5 ] ], B ); <A morphism in FinSets> gap> f3 := MapOfFinSets( A, [ [ 1, 4 ], [ 2, 2 ], [ 3, 3 ], [ 4, 8 ] ], B ); <A morphism in FinSets> gap> C1 := Coequalizer( [ f1, f3 ] ); <An object in FinSets> gap> Display( AsList( C1 ) ); [ [ 1, 4 ], [ 2 ], [ 3 ], [ 5 ], [ 6 ], [ 7 ], [ 8 ] ] gap> C2 := Coequalizer( [ f1, f2, f3 ] ); <An object in FinSets> gap> Display( AsList( C2 ) ); [ [ 1, 2, 3, 8, 5, 4 ], [ 6 ], [ 7 ] ] gap> S := FinSet( [ 1 .. 5 ] ); <An object in FinSets> gap> T := FinSet( [ 1 .. 4 ] ); <An object in FinSets> gap> f := MapOfFinSets( S, [ [1,2], [2,4], [3,4], [4,3], [5,4] ], T ); <A morphism in FinSets> gap> g := MapOfFinSets( S, [ [1,2], [2,3], [3,4], [4,3], [5,4] ], T ); <A morphism in FinSets> gap> C := Coequalizer( f, g ); <An object in FinSets> gap> Display( C ); [ [ 1 ], [ 2 ], [ 4, 3 ] ] gap> S := FinSet( [ 1, 2, 3, 4, 5 ] ); <An object in FinSets> gap> T := FinSet( [ 1, 2, 3, 4 ] ); <An object in FinSets> gap> G_f := [ [ 1, 3 ], [ 2, 4 ], [ 3, 4 ], [ 4, 2 ], [ 5, 4 ] ];; gap> f := MapOfFinSets( S, G_f, T ); <A morphism in FinSets> gap> G_g := [ [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 2 ], [ 5, 4 ] ];; gap> g := MapOfFinSets( S, G_g, T ); <A morphism in FinSets> gap> D := [ f, g ];; gap> C := Coequalizer( D ); <An object in FinSets> gap> Display( AsList( C ) ); [ [ 1 ], [ 2 ], [ 3, 4 ] ] gap> pi := ProjectionOntoCoequalizer( D ); <An epimorphism in FinSets> gap> Display( AsList( pi ) ); [ [ 1, [ 1 ] ], [ 2, [ 2 ] ], [ 3, [ 3, 4 ] ], [ 4, [ 3, 4 ] ] ] gap> mu := MorphismFromSourceToCoequalizer( D );; gap> PreCompose( f, pi ) = mu; true gap> G_tau := [ [ 1, 2 ], [ 2, 1 ], [ 3, 2 ], [ 4, 2 ] ];; gap> tau := MapOfFinSets( T, G_tau, FinSet( [ 1, 2 ] ) ); <A morphism in FinSets> gap> phi := UniversalMorphismFromCoequalizer( D, tau ); <A morphism in FinSets> gap> Display( AsList( phi ) ); [ [ [ 1 ], 2 ], [ [ 2 ], 1 ], [ [ 3, 4 ], 2 ] ] gap> PreCompose( pi, phi ) = tau; true gap> S := FinSet( [ 1, 2, 3, 4, 5 ] ); <An object in FinSets> gap> T := FinSet( [ 1, 2, 3, 4 ] ); <An object in FinSets> gap> G_f := [ [ 1, 2 ], [ 2, 3 ], [ 3, 3 ], [ 4, 2 ], [ 5, 4 ] ];; gap> f := MapOfFinSets( S, G_f, T ); <A morphism in FinSets> gap> G_g := [ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 2 ], [ 5, 4 ] ];; gap> g := MapOfFinSets( S, G_g, T ); <A morphism in FinSets> gap> D := [ f, g ];; gap> C := Coequalizer( D ); <An object in FinSets> gap> Display( AsList( C ) ); [ [ 1 ], [ 2, 3 ], [ 4 ] ] gap> pi := ProjectionOntoCoequalizer( D ); <An epimorphism in FinSets> gap> Display( AsList( pi ) ); [ [ 1, [ 1 ] ], [ 2, [ 2, 3 ] ], [ 3, [ 2, 3 ] ], [ 4, [ 4 ] ] ] gap> PreCompose( f, pi ) = PreCompose( g, pi ); true gap> mu := MorphismFromSourceToCoequalizer( D );; gap> PreCompose( f, pi ) = mu; true gap> G_tau := [ [ 1, 1 ], [ 2, 2 ], [ 3, 2 ], [ 4, 1 ] ];; gap> tau := MapOfFinSets( T, G_tau, FinSet( [ 1, 2 ] ) ); <A morphism in FinSets> gap> phi := UniversalMorphismFromCoequalizer( D, tau ); <A morphism in FinSets> gap> Display( AsList( phi ) ); [ [ [ 1 ], 1 ], [ [ 2, 3 ], 2 ], [ [ 4 ], 1 ] ] gap> PreCompose( pi, phi ) = tau; true gap> A := FinSet( [ "A" ] ); <An object in FinSets> gap> B := FinSet( [ "B" ] ); <An object in FinSets> gap> M := FinSetNC( [ A, B ] ); <An object in FinSets> gap> f := MapOfFinSetsNC( M, [ [ A, A ], [ B, A ] ], M ); <A morphism in FinSets> gap> g := IdentityMorphism( M ); <An identity morphism in FinSets> gap> C := Coequalizer( [ f, g ] ); <An object in FinSets> gap> Length( C ); 1 gap> Length( AsList( C )[ 1 ] ); 2 gap> Display( AsList( C )[ 1 ][ 1 ] ); [ "A" ] gap> Display( AsList( C )[ 1 ][ 2 ] ); [ "B" ]
gap> LoadPackage( "FinSetsForCAP", false ); true gap> M := FinSet( [ 1 .. 5 ] ); <An object in FinSets> gap> N1 := FinSet( [ 1, 2, 4 ] ); <An object in FinSets> gap> iota1 := EmbeddingOfFinSets( N1, M ); <A monomorphism in FinSets> gap> N2 := FinSet( [ 2, 3 ] ); <An object in FinSets> gap> iota2 := EmbeddingOfFinSets( N2, M ); <A monomorphism in FinSets> gap> D := [ iota1, iota2 ];; gap> int := FiberProduct( D ); <An object in FinSets> gap> Display( int ); [ [ 2, 2 ] ] gap> pi1 := ProjectionInFactorOfFiberProduct( D, 1 ); <A monomorphism in FinSets> gap> pi2 := ProjectionInFactorOfFiberProduct( D, 2 ); <A monomorphism in FinSets> gap> UU := Pushout( pi1, pi2 ); <An object in FinSets> gap> Display( UU ); [ [ [ 1, 1 ] ], [ [ 1, 2 ], [ 2, 2 ] ], [ [ 1, 4 ] ], [ [ 2, 3 ] ] ] gap> iota := UniversalMorphismFromPushout( [ pi1, pi2 ], [ iota1, iota2 ] ); <A morphism in FinSets> gap> U := ImageObject( iota ); <An object in FinSets> gap> Display( U ); [ 1, 2, 4, 3 ] gap> UnionOfFinSets( FinSets, [ N1, N2 ] ) = U; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( [ 1 .. 3 ] ); <An object in FinSets> gap> R := FinSet( [ 1 .. 2 ] ); <An object in FinSets> gap> f := MapOfFinSets( S, [ [1,2],[2,2],[3,1] ], R ); <A morphism in FinSets> gap> IsWellDefined( f ); true gap> T := TerminalObject( FinSets ); <An object in FinSets> gap> IsTerminal( T ); true gap> lf := CartesianLambdaIntroduction( f ); <A split monomorphism in FinSets> gap> Source( lf ) = T; true gap> Length( Range( lf ) ); 8 gap> lf( T[1] ) = f; true gap> elf := CartesianLambdaElimination( S, R, lf ); <A morphism in FinSets> gap> elf = f; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( [ 1 .. 3 ] ); <An object in FinSets> gap> n := FinSet( [ 1 .. 4 ] ); <An object in FinSets> gap> f := MapOfFinSets( m, [ [ 1, 2 ], [ 2, 2 ], [ 3, 1 ] ], m ); <A morphism in FinSets> gap> g := MapOfFinSets( n, [ [ 1, 3 ], [ 2, 2 ], [ 3, 1 ], [ 4, 2 ] ], m ); <A morphism in FinSets> gap> IsLiftable( f, g ); true gap> chi := Lift( f, g ); <A morphism in FinSets> gap> Display( chi ); [ [ 1 .. 3 ], [ [ 1, 2 ], [ 2, 2 ], [ 3, 3 ] ], [ 1 .. 4 ] ] gap> PreCompose( Lift( f, g ), g ) = f; true gap> IsLiftable( g, f ); false gap> k := FinSet( [ 1 .. 100 ] ); <An object in FinSets> gap> h := ListWithIdenticalEntries( Length( k ) - 3, 3 );; gap> h := Concatenation( h, [ 2, 1, 2 ] );; gap> h := MapOfFinSets( k, List( k, i -> [ i, h[i] ] ), m ); <A morphism in FinSets> gap> IsLiftable( f, h ); true gap> IsLiftable( h, f ); false
gap> LoadPackage( "FinSetsForCAP" ); true gap> a := FinSet( [ 1, 2, 3 ] ); <An object in FinSets> gap> sa := SingletonMorphism( a ); <A monomorphism in FinSets> gap> sa = LowerSegmentOfRelation( a, a, CartesianDiagonal( a, 2 ) ); true gap> sa = UpperSegmentOfRelation( a, a, CartesianDiagonal( a, 2 ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> M := FinSet( [ 1 .. 5 ] );; gap> N := FinSet( [ 1, 2, 4 ] );; gap> P := FinSet( [ 1, 4, 8, 9 ] );; gap> G_f := [ [ 1, 1 ], [ 2, 2 ], [ 3, 1 ], [ 4, 2 ], [ 5, 4 ] ];; gap> f := MapOfFinSets( M, G_f, N );; gap> IsWellDefined( f ); true gap> G_g := [ [ 1, 4 ], [ 2, 4 ], [ 3, 2 ], [ 4, 2 ], [ 5, 1 ] ];; gap> g := MapOfFinSets( M, G_g, N );; gap> IsWellDefined( g ); true gap> DirectProduct( M, N );; gap> DirectProductOnMorphisms( f, g );; gap> CartesianAssociatorLeftToRight( M, N, P );; gap> CartesianAssociatorRightToLeft( M, N, P );; gap> TerminalObject( FinSets );; gap> CartesianLeftUnitor( M );; gap> CartesianLeftUnitorInverse( M );; gap> CartesianRightUnitor( M );; gap> CartesianRightUnitorInverse( M );; gap> CartesianBraiding( M, N );; gap> CartesianBraidingInverse( M, N );; gap> exp := ExponentialOnObjects( M, N );; gap> ExponentialOnObjects( FinSet( [ 1 ] ), exp );; gap> ExponentialOnMorphisms( f, g );; gap> CartesianRightEvaluationMorphism( M, N );; gap> CartesianRightCoevaluationMorphism( N, M );; gap> DirectProductToExponentialRightAdjunctMorphism( M, N, > UniversalMorphismIntoTerminalObject( DirectProduct( M, N ) ) > );; gap> ExponentialToDirectProductRightAdjunctMorphism( M, N, > UniversalMorphismFromInitialObject( ExponentialOnObjects( M, N ) ) > );; gap> CartesianLeftEvaluationMorphism( M, N );; gap> CartesianLeftCoevaluationMorphism( N, M );; gap> DirectProductToExponentialLeftAdjunctMorphism( M, N, > UniversalMorphismIntoTerminalObject( DirectProduct( M, N ) ) > );; gap> ExponentialToDirectProductLeftAdjunctMorphism( M, N, > UniversalMorphismFromInitialObject( ExponentialOnObjects( M, N ) ) > );; gap> M := FinSet( [ 1, 2 ] );; gap> N := FinSet( [ "a", "b" ] );; gap> P := FinSet( [ "*" ] );; gap> Q := FinSet( [ "§", "&" ] );; gap> CartesianPreComposeMorphism( M, N, P );; gap> CartesianPostComposeMorphism( M, N, P );; gap> DirectProductExponentialCompatibilityMorphism( [ M, N, P, Q ] );;
gap> LoadPackage( "FinSetsForCAP", false ); true gap> Display( SubobjectClassifier( FinSets ) ); [ "true", "false" ] gap> Display( TruthMorphismOfTrue( FinSets ) ); [ [ [ ] ], [ [ [ ], "true" ] ], [ "true", "false" ] ] gap> Display( TruthMorphismOfFalse( FinSets ) ); [ [ [ ] ], [ [ [ ], "false" ] ], [ "true", "false" ] ] gap> Display( TruthMorphismOfNot( FinSets ) ); [ [ "true", "false" ],\ [ [ "false", "true" ], [ "true", "false" ] ],\ [ "true", "false" ] ] gap> Display( CartesianSquareOfSubobjectClassifier( FinSets ) ); [ [ "true", "true" ], [ "false", "true" ],\ [ "true", "false" ], [ "false", "false" ] ] gap> Display( TruthMorphismOfAnd( FinSets ) ); [ [ [ "true", "true" ], [ "false", "true" ],\ [ "true", "false" ], [ "false", "false" ] ],\ [ [ [ "false", "false" ], "false" ], [ [ "false", "true" ], "false" ],\ [ [ "true", "false" ], "false" ], [ [ "true", "true" ], "true" ] ],\ [ "true", "false" ] ] gap> Display( TruthMorphismOfOr( FinSets ) ); [ [ [ "true", "true" ], [ "false", "true" ],\ [ "true", "false" ], [ "false", "false" ] ],\ [ [ [ "false", "false" ], "false" ], [ [ "false", "true" ], "true" ],\ [ [ "true", "false" ], "true" ], [ [ "true", "true" ], "true" ] ],\ [ "true", "false" ] ] gap> Display( TruthMorphismOfImplies( FinSets ) ); [ [ [ "true", "true" ], [ "false", "true" ],\ [ "true", "false" ], [ "false", "false" ] ],\ [ [ [ "false", "false" ], "true" ], [ [ "false", "true" ], "true" ],\ [ [ "true", "false" ], "false" ], [ [ "true", "true" ], "true" ] ],\ [ "true", "false" ] ] gap> S := FinSet( [ 1, 2, 3, 4, 5 ] ); <An object in FinSets> gap> A := FinSet( [ 1, 5 ] ); <An object in FinSets> gap> m := EmbeddingOfFinSets( A, S ); <A monomorphism in FinSets> gap> Display( ClassifyingMorphismOfSubobject( m ) ); [ [ 1, 2, 3, 4, 5 ], [ [ 1, "true" ], [ 2, "false" ], [ 3, "false" ],\ [ 4, "false" ], [ 5, "true" ] ], [ "true", "false" ] ]
gap> LoadPackage( "FinSetsForCAP", false ); true gap> M := FinSet( [ 1 .. 7 ] ); <An object in FinSets> gap> N := FinSet( [ 2, 3, 5 ] ); <An object in FinSets> gap> iotaN := EmbeddingOfFinSets( N, M ); <A monomorphism in FinSets> gap> NC := PseudoComplementSubobject( iotaN ); <An object in FinSets> gap> Display( NC ); [ [ 1, [ ] ], [ 4, [ ] ], [ 6, [ ] ], [ 7, [ ] ] ] gap> tauN := EmbeddingOfPseudoComplementSubobject( iotaN ); <A monomorphism in FinSets> gap> Nc := ImageObject( tauN ); <An object in FinSets> gap> Display( Nc ); [ 1, 4, 6, 7 ] gap> L := FinSet( [ 2, 4, 5, 7 ] ); <An object in FinSets> gap> iotaL := EmbeddingOfFinSets( L, M ); <A monomorphism in FinSets> gap> NIL := IntersectionSubobject( iotaN, iotaL ); <An object in FinSets> gap> Display( NIL ); [ [ 2, 2 ], [ 5, 5 ] ] gap> iotaNiL := EmbeddingOfIntersectionSubobject( iotaN, iotaL ); <A monomorphism in FinSets> gap> NiL := ImageObject( iotaNiL ); <An object in FinSets> gap> Display( NiL ); [ 2, 5 ] gap> NUL := UnionSubobject( iotaN, iotaL ); <An object in FinSets> gap> Display( NUL ); [ 2, 3, 5, 4, 7 ] gap> iotaNuL := EmbeddingOfUnionSubobject( iotaN, iotaL ); <A monomorphism in FinSets> gap> NuL := ImageObject( iotaNuL ); <An object in FinSets> gap> Display( NuL ); [ 2, 3, 5, 4, 7 ] gap> NPL := RelativePseudoComplementSubobject( iotaN, iotaL ); <An object in FinSets> gap> Display( NPL ); [ [ 1, [ ] ], [ 2, [ ] ], [ 4, [ ] ], [ 5, [ ] ], [ 6, [ ] ], [ 7, [ ] ]\ ] gap> iotaNpL := EmbeddingOfRelativePseudoComplementSubobject( iotaN, iotaL ); <A monomorphism in FinSets> gap> NpL := ImageObject( iotaNpL ); <An object in FinSets> gap> Display( NpL ); [ 1, 2, 4, 5, 6, 7 ]
Define two composable monos \(K \stackrel{l}{\hookrightarrow} L \stackrel{m}{\hookrightarrow} G\) in FinSets:
gap> LoadPackage( "FinSetsForCAP", false ); true gap> K := FinSet( [ 2, 3, 5 ] );; gap> Display( K ); [ 2, 3, 5 ] gap> L := FinSet( [ 0, 1, 2, 3, 5, 6 ] );; gap> Display( L ); [ 0, 1, 2, 3, 5, 6 ] gap> l := EmbeddingOfFinSets( K, L ); <A monomorphism in FinSets> gap> G := FinSet( [ 0, 1, 2, 3, 4, 5, 6, 10 ] );; gap> Display( G ); [ 0, 1, 2, 3, 4, 5, 6, 10 ] gap> m := EmbeddingOfFinSets( L, G ); <A monomorphism in FinSets>
Now we compute the pushout complement \(D \stackrel{c}{\hookrightarrow} G\) of \(K \stackrel{l}{\hookrightarrow} L \stackrel{m}{\hookrightarrow} G\):
gap> LoadPackage( "FinSetsForCAP", false ); true gap> HasPushoutComplement( l, m ); true gap> c := PushoutComplement( l, m ); <A monomorphism in FinSets> gap> D := Source( c );; gap> Display( D ); [ 2, 3, 4, 5, 10 ]
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