‣ IsFinQuiver( q ) | ( filter ) |
Returns: true or false
The GAP category of quivers.
‣ IsFinQuiverObject( v ) | ( filter ) |
Returns: true or false
The GAP category of objects in a quiver.
‣ IsFinQuiverMorphism( alpha ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a quiver.
‣ FinQuiver( q_datum ) | ( function ) |
Returns: a CAP quiver
Construct a CAP quiver out of the q_datum which is specified as follows:
q_datum[1] is a string representing the name of the quiver.
q_datum[2] is a list consisting of two entries: the number of objects and the list of labels for these objects. This list might contain an optional third entry for the list of LaTeX-strings for the objects.
q_datum[3] is a list consisting of four entries: the number of morphisms, the list of source-indices, the list of range-indices and the list of labels for these morphisms. This list might contain an optional fifth entry for the list of LaTeX-strings for the morphisms.
gap> LoadPackage( "FpCategories", false ); true gap> str := > "q(0..5)[x:0->0,s:0->1,a:1->2,c:1->3,e:1->4,b:2->4,d:3->4,t:4->5,y:5->5]";; gap> q := FinQuiver( str ); FinQuiver( "q(0,1,2,3,4,5)[x:0-≻0,s:0-≻1,a:1-≻2,c:1-≻3,e:1-≻4,b:2-≻4,d:3-≻4, t:4-≻5,y:5-≻5]" ) gap> QuiverName( q ); "q" gap> NumberOfObjects( q ); 6 gap> LabelsOfObjects( q ); [ "0", "1", "2", "3", "4", "5" ] gap> SetOfObjects( q ); [ (0), (1), (2), (3), (4), (5) ] gap> o := q.0; (0) gap> ObjectIndex( o ); 1 gap> IsIdenticalObj( o, ObjectConstructor( q, 1 ) ); true gap> ObjectLabel( o ); "0" gap> LaTeXOutput( o ); "0" gap> NumberOfMorphisms( q ); 9 gap> LabelsOfMorphisms( q ); [ "x", "s", "a", "c", "e", "b", "d", "t", "y" ] gap> IndicesOfSources( q ); [ 1, 1, 2, 2, 2, 3, 4, 5, 6 ] gap> IndicesOfTargets( q ); [ 1, 2, 3, 4, 5, 5, 5, 6, 6 ] gap> SetOfMorphisms( q ); [ x:(0) -≻ (0), s:(0) -≻ (1), a:(1) -≻ (2), c:(1) -≻ (3), e:(1) -≻ (4), b:(2) -≻ (4), d:(3) -≻ (4), t:(4) -≻ (5), y:(5) -≻ (5) ] gap> m := q.y; y:(5) -≻ (5) gap> MorphismIndex( m ); 9 gap> IsIdenticalObj( m, MorphismConstructor( q, q.("5"), 9, q.("5") ) ); true gap> MorphismLabel( m ); "y" gap> LaTeXOutput( m ); "{5}-\\left({y}\\right)\\rightarrow{5}" gap> MorphismsOfExternalHom( q.("0"), q.("0") ); [ x:(0) -≻ (0) ] gap> MorphismsOfExternalHom( q.("0"), q.("1") ); [ s:(0) -≻ (1) ] gap> q_op := OppositeQuiver( q ); FinQuiver( "q_op(0,1,2,3,4,5)[x:0-≻0,s:1-≻0,a:2-≻1,c:3-≻1,e:4-≻1,b:4-≻2, d:4-≻3,t:5-≻4,y:5-≻5]" ) gap> MorphismsOfExternalHom( q_op.1, q_op.0 ); [ s:(1) -≻ (0) ] gap> q_op_x_q := TensorProductOfFinQuivers( q_op, q );; gap> NumberOfObjects( q_op_x_q ); 36 gap> NumberOfMorphisms( q_op_x_q ); 108
gap> q1 := RandomFinQuiver( 2, 4, false );; # without loops gap> q2 := RandomFinQuiver( 2, 4, true );; # with loops gap> Assert( 0, NumberOfObjects( q1 ) = 2 ); gap> Assert( 0, NumberOfMorphisms( q1 ) = 4 ); gap> Assert( 0, Length( Filtered( SetOfMorphisms( q1 ), IsEndomorphism ) ) = 0 ); gap> Assert( 0, Length( Filtered( SetOfMorphisms( q2 ), IsEndomorphism ) ) > 0 ); gap> q3 := FinQuiver( "q(3)[a:1->2,b:2->3]" ); FinQuiver( "q(1,2,3)[a:1-≻2,b:2-≻3]" ) gap> q4 := FinQuiver( "q(4..6)[a:4->5,b:5->6]" ); FinQuiver( "q(4,5,6)[a:4-≻5,b:5-≻6]" )
‣ ObjectConstructor( C, i ) | ( operation ) |
Returns: a CAP category object
The input is a CAP quiver \(q\) and a positive integer i. The output is the i'th object in q.
‣ MorphismConstructor( C, s, i, t ) | ( operation ) |
Returns: a CAP category object
The input is a CAP quiver \(q\), an object s, a positive integer i and an object t. The output is the i'th morphism in q (whose source is s and target is t).
‣ RandomFinQuiver( nr_objs, nr_mors, has_loops ) | ( function ) |
Returns: a CAP quiver
The outout is a random quiver consisting of nr_objs objects and nr_mors morphisms and has loops only if has_loops = true.
‣ TensorProductOfFinQuivers( q1, q2 ) | ( operation ) |
Returns: a CAP quiver
Returns the tensor product of q1 and q2.
‣ QuiverDatum( q ) | ( attribute ) |
Returns: a list
Returns the quiver datum of the quiver q.
‣ QuiverName( q ) | ( attribute ) |
Returns: a list
Returns the name of the quiver q.
‣ NumberOfObjects( q ) | ( attribute ) |
Returns: a positive integer
Returns the number of objects in the quiver q.
‣ LabelsOfObjects( q ) | ( attribute ) |
Returns: a list
Returns the labels of all objects in the quiver q.
‣ LaTeXStringsOfObjects( q ) | ( attribute ) |
Returns: a list
Returns the LaTeX-strings of all objects in the quiver q.
‣ NumberOfMorphisms( q ) | ( attribute ) |
Returns: a positive integer
Returns the number of morphisms in the quiver q.
‣ LabelsOfMorphisms( q ) | ( attribute ) |
Returns: a list
Returns the labels of all morphisms in the quiver q.
‣ LaTeXStringsOfMorphisms( q ) | ( attribute ) |
Returns: a list
Returns the LaTeX-strings of all morphisms in the quiver q.
‣ IndicesOfSources( q ) | ( attribute ) |
Returns: a list
Returns the indices of sources of all morphisms in the quiver q.
‣ IndicesOfTargets( q ) | ( attribute ) |
Returns: a list
Returns the indices of targets of all morphisms in the quiver q.
‣ IndicesPairsOfCompatibleMorphisms( q ) | ( attribute ) |
Returns: a list
Returns the indices of all paris of morphisms where the target of the first morphism equals the source of the second morphism.
‣ SetOfObjects( q ) | ( attribute ) |
Returns: a list of CAP quiver objects
Returns the set of objects of q.
‣ SetOfMorphisms( q ) | ( attribute ) |
Returns: a list of CAP quiver morphisms
Returns the set of morphisms of q.
‣ ExternalHoms( q ) | ( attribute ) |
Returns: a list of CAP quiver morphisms
Returns a list of lists where the entry at \([i,j]\) is the set of morphism from the \(i\)'th object to the \(j\)'th object in q.
‣ OppositeQuiver( q ) | ( attribute ) |
Returns: CAP quiver
Returns the opposite quiver of q.
‣ ObjectIndex( v ) | ( attribute ) |
Returns: a positive integer
Returns the index of the object v.
‣ ObjectLabel( v ) | ( attribute ) |
Returns: a string
Returns the label of the object v.
‣ UnderlyingFinQuiver( v ) | ( attribute ) |
Returns: a CAP quiver
Returns the quiver containing the object v.
‣ MorphismIndex( alpha ) | ( attribute ) |
Returns: a positive integer
Returns the index of the morphism alpha.
‣ MorphismLabel( alpha ) | ( attribute ) |
Returns: a string
Returns the label of the morphism alpha.
‣ UnderlyingFinQuiver( alpha ) | ( attribute ) |
Returns: a CAP quiver
Returns the quiver containing the morphism alpha.
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