‣ IsFinitelyPresentedLinearCategory( A ) | ( property ) |
Returns: true or false
Is the linear category A finitely presented. This property is true by construction for linear closures of path categories and their quotients, and for algebroids defined by path algebras or their quotients.
‣ IsFpLinearCategory( A ) | ( category ) |
Returns: true or false
The filter for finitely presented linear CAP categories.
‣ IsObjectInFpLinearCategory( obj ) | ( category ) |
Returns: true or false
The filter for objects in finitely presented linear CAP categories.
‣ IsMorphismInFpLinearCategory( mor ) | ( category ) |
Returns: true or false
The filter for morphisms in finitely presented linear CAP categories.
‣ IsLinearClosureOfPathCategory( kC ) | ( category ) |
Returns: true or false
The filter for linear closures of path categories.
‣ IsObjectInLinearClosureOfPathCategory( obj ) | ( category ) |
Returns: true or false
The filter for objects in linear closures of path categories.
‣ IsMorphismInLinearClosureOfPathCategory( mor ) | ( category ) |
Returns: true or false
The filter for morphisms in linear closures of path categories.
‣ IsLinearClosureOfQuotientOfPathCategory( kC ) | ( category ) |
Returns: true or false
The filter for linear closures of quotients of path categories.
‣ IsObjectInLinearClosureOfQuotientOfPathCategory( obj ) | ( category ) |
Returns: true or false
The filter for objects in linear closures of quotients of path categories.
‣ IsMorphismInLinearClosureOfQuotientOfPathCategory( mor ) | ( category ) |
Returns: true or false
The filter for morphisms in linear closures of quotients of path categories.
‣ LinearClosure( k, C ) | ( operation ) |
Returns: a CAP category
Returns the \(k\)-linear closure category of C.
‣ IsQuotientCategoryOfLinearClosureOfPathCategory( kC ) | ( category ) |
Returns: true or false
The filter for quotient categories of linear closures of path categories.
‣ IsObjectInQuotientCategoryOfLinearClosureOfPathCategory( obj ) | ( category ) |
Returns: true or false
The filter for objects in quotient categories of linear closures of path categories.
‣ IsMorphismInQuotientCategoryOfLinearClosureOfPathCategory( mor ) | ( category ) |
Returns: true or false
The filter for morphisms in quotient categories of linear closures of path categories.
‣ IsQuotientCategoryOfLinearClosureOfQuotientOfPathCategory( kC ) | ( category ) |
Returns: true or false
The filter for quotient categories of linear closures of quotients of path categories.
‣ IsMorphismInQuotientCategoryOfLinearClosureOfQuotientOfPathCategory( mor ) | ( category ) |
Returns: true or false
The filter for objects in quotient categories of linear closures of quotients of path categories. The filter for morphisms in quotient categories of linear closures of quotients of path categories.
‣ QuotientCategory( kC, relations ) | ( operation ) |
Returns: a list of morphisms
Returns a the quotient category of kC modulo the two-sided ideal generated by relations. This method requires the Groebner basis of relations to be a finite set.
gap> LoadPackage( "FpLinearCategories", 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> k := HomalgFieldOfRationals( );; gap> C := PathCategory( q : admissible_order := "Dp" ); PathCategory( 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> kC := LinearClosure( k, C ); Q-LinearClosure( PathCategory( 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> IsAdmissibleAlgebroid( kC ); false gap> rels := [ kC.x^10 - kC.x^5, kC.abt - kC.et, kC.y^10 - kC.y^5, kC.x^5, kC.y^5 ];; gap> Perform( rels, Display ); 1*x^10 + (-1)*x^5:(0) → (0) 1*a⋅b⋅t + (-1)*e⋅t:(1) → (5) 1*y^10 + (-1)*y^5:(5) → (5) 1*x^5:(0) → (0) 1*y^5:(5) → (5) gap> quo_kC := QuotientCategory( kC, rels ); Q-LinearClosure( PathCategory( 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]" ) ) ) / [ 1*x^10 + (-1)*x^5, 1*a⋅b⋅t + (-1)*e⋅t, 1*y^10 + (-1)*y^5, ... ] gap> IsAdmissibleAlgebroid( quo_kC ); true gap> HomStructure( quo_kC.("0"), quo_kC.("5") ); <A row module over Q of rank 50> gap> A := AlgebroidFromDataTables( quo_kC ); Q-algebroid( {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] ) defined by 6 objects and 9 generating morphisms gap> HomStructure( A.("0"), A.("5") ); <A row module over Q of rank 50> gap> quo_C := C / [ [ C.x^10, C.x^5 ], [ C.abt, C.et ], [ C.y^10, C.y^5 ] ]; PathCategory( 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]" ) ) / [ x^10 = x^5, a⋅b⋅t = e⋅t, y^10 = y^5 ] gap> k_quo_C := k[quo_C]; Q-LinearClosure( PathCategory( 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]" ) ) / [ x^10 = x^5, a⋅b⋅t = e⋅t, y^10 = y^5 ] ) gap> quo_k_quo_C := k_quo_C / [ k_quo_C.x^5, k_quo_C.y^5 ]; Q-LinearClosure( PathCategory( 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]" ) ) / [ x^10 = x^5, a⋅b⋅t = e⋅t, y^10 = y^5 ] ) / [ 1*[x^5], 1*[y^5] ] gap> HomStructure( quo_k_quo_C.("0"), quo_k_quo_C.("5") ); <A row module over Q of rank 50> gap> Dimension( quo_k_quo_C ); 126 gap> IsAdmissibleAlgebroid( quo_k_quo_C ); true gap> ModelingCategory( quo_k_quo_C ); Q-LinearClosure( PathCategory( 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]" ) ) ) / [ 1*x^10 + (-1)*x^5, 1*a⋅b⋅t + (-1)*e⋅t, 1*y^10 + (-1)*y^5, ... ] gap> B := AlgebroidFromDataTables( quo_k_quo_C ); Q-algebroid( {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] ) defined by 6 objects and 9 generating morphisms gap> HomStructure( B.("0"), B.("5") ); <A row module over Q of rank 50>
‣ GroebnerBasis( kC, relations ) | ( operation ) |
Returns: a list of morphisms
Returns the Groebner basis of the two-sided ideal generated by relations.
‣ ReducedGroebnerBasis( kC, relations ) | ( operation ) |
Returns: a list of morphisms
Returns a reduced Groebner basis of the two-sided ideal generated by relations.
gap> LoadPackage( "FpLinearCategories", false ); true gap> k := HomalgFieldOfRationals( );; gap> q := FinQuiver( "q(o)[x:o->o,y:o->o]" ); FinQuiver( "q(o)[x:o→o,y:o→o]" ) gap> C := PathCategory( q ); PathCategory( FinQuiver( "q(o)[x:o→o,y:o→o]" ) ) gap> kC := k[C]; Q-LinearClosure( PathCategory( FinQuiver( "q(o)[x:o→o,y:o→o]" ) ) ) gap> x := kC.x; 1*x:(o) → (o) gap> y := kC.y; 1*y:(o) → (o) gap> rels := [ x*y-y*x, (x^2+y^2)*(x+x*y), (x^2+y^2)*(y^2+x^3) ];; gap> Perform( rels, Display ); (-1)*y⋅x + 1*x⋅y:(o) → (o) 1*x^3⋅y + 1*y^2⋅x⋅y + 1*x^3 + 1*y^2⋅x:(o) → (o) 1*x^5 + 1*y^2⋅x^3 + 1*x^2⋅y^2 + 1*y^4:(o) → (o) gap> gb := ReducedGroebnerBasis( kC, rels );; gap> Perform( gb, Display ); (-1)*y⋅x + 1*x⋅y:(o) → (o) 1*x^3⋅y + 1*x⋅y^3 + 1*x^3 + 1*x⋅y^2:(o) → (o) 1*x^5 + (-1)*x⋅y^4 + 1*x^2⋅y^2 + 1*y^4 + 1*x^3 + 1*x⋅y^2:(o) → (o) 1*x^2⋅y^3 + 1*y^5 + 1*x^2⋅y^2 + 1*y^4:(o) → (o) gap> f := (x-y)*gb[1] + (x^2-y)*gb[2] + y^3*gb[3] + (x-y^3)*gb[4]; 1*y^3⋅x^5 + (-1)*y^3⋅x^2⋅y^3 + (-1)*y^3⋅x⋅y^4 + (-1)*y^8 + 1*y^3⋅x^3 + 1*x^5⋅y + 1*y^3⋅x⋅y^2 + 2*x^3⋅y^3 + 1*x⋅y^5 + 1*x^5 + (-1)*y⋅x^3⋅y + 2*x^3⋅y^2 + (-1)*y⋅x⋅y^3 + 1*x⋅y^4 + (-1)*y⋅x^3 + (-1)*y⋅x⋅y^2 + (-1)*x⋅y⋅x + 1*y^2⋅x + 1*x^2⋅y + (-1)*y⋅x⋅y:(o) → (o) gap> ReductionOfMorphism( kC, f, rels ); (-1)*x^2⋅y^6 + (-1)*y^8 + 1*x^2⋅y^2 + 1*y^4:(o) → (o) gap> ReductionOfMorphism( kC, f, gb ); 0:(o) → (o)
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