GAP (Algebroids) - Chapter 10: Category of finitely presented associative unital algebras

The input is a category of finitely presented associative unital k-algebras. The output is the underlying commutative ring k of coefficients.

10.1-2 CoefficientsRing

The input is a finitely presented associative unital k-algebras. The output is the underlying commutative ring k of coefficients.

10.1-3 DefiningSeptupleOfFinitelyPresentedAlgebra

10.1-4 ListOfImages

10.1-5 NrGenerators

The input is a finitely presented associative unital algebra. The output is the number of generators.

10.1-6 Generators

The input is a finitely presented associative unital algebra. The output is the list of generators of the underlying finitely presented associative unital algebra.

10.1-7 AssociatedLinearClosureOfPathCategory

The input is a finitely presented associative unital k-algebra. The output is the corresponding ambient free k-algebra as a k-linear closure of a path category of loops on a single vertex.

10.1-8 DefiningRelations

The input is a finitely presented associative unital algebra. The output its list of defining relations.

10.1-9 GroebnerBasisOfDefiningRelations

The input is a finitely presented associative unital algebra. The output is the Gröbner basis of its list of defining relations.

10.1-10 AssociatedQuotientCategoryOfLinearClosureOfPathCategory

The input is a finitely presented associative unital k-algebra. The output is the corresponding underlying finitely presented associative unital k-algebra as a quotient of a k-linear closure of a path category of loops on a single vertex.

10.1-11 Dimension

The input is a finitely presented associative unital k-algebra. The output its k-dimension.

10.1-12 AmbientAlgebra

The input is a finitely presented associative unital algebra. The output is the corresponding ambient free algebra.

10.1-13 AssociatedFunctorOfLinearClosuresOfPathCategoriesData

10.2 Constructors

10.2-1 CategoryOfFpAlgebras

The input is a commutative ring k. The output is the category of finitely presented associative unital algebras over k.

gap> LoadPackage( "Algebroids", false );
true
gap> F2 := HomalgRingOfIntegersInSingular( 2 );
GF(2)
gap> Alg_F2 := CategoryOfFpAlgebras( F2 );
CategoryOfFpAlgebras( GF(2) )
gap> Display( Alg_F2 );
A CAP category with name CategoryOfFpAlgebras( GF(2) ):

31 primitive operations were used to derive 130 operations for this category \
which algorithmically
* IsFiniteCocompleteCategory
* IsBicartesianCategory
* IsSymmetricMonoidalCategory
gap> T := TerminalObject( Alg_F2 );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Display( T );
GF(2) / [ 1*id(o) ]
gap> iota := UniversalMorphismFromInitialObject( T );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> I := Source( iota );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Display( I );
GF(2) / [  ]
gap> Display( iota );
GF(2) / [ 1*id(o) ]
 ^
 |
[  ]
 |
GF(2) / [  ]
gap> K := TensorUnit( Alg_F2 );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Display( K );
GF(2) / [  ]
gap> K = I;
true
gap> q1 := FinQuiver( "q(o)[x:o->o,y:o->o]" );
FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" )
gap> P1 := PathCategory( q1 );
PathCategory( FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) )
gap> L1 := F2[P1];
GF(2)-LinearClosure( PathCategory( \
FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) ) )
gap> relations1 := [ L1.x^2 - L1.x,  L1.y^3 - L1.y,  L1.xy - L1.y * (L1.x+L1.id_o) ];
[ 1*x^2 + 1*x:(o) -≻ (o), 1*y^3 + 1*y:(o) -≻ (o),\
  1*y⋅x + 1*x⋅y + 1*y:(o) -≻ (o) ]
gap> Q1 := L1 / relations1;
GF(2)-LinearClosure( PathCategory( FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) ) ) / \
[ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
gap> Dimension( Q1 );
6
gap> AQ1 := AssociatedAlgebroid( Q1 );
Algebra( GF(2), FreeCategory( RightQuiver( "q(o)[x:o->o,y:o->o]" ) ) ) / \
relations
gap> Dimension( AQ1 );
6
gap> A1 := Q1 / Alg_F2;
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> A1 = ( L1 / Alg_F2 ) / relations1;
true
gap> Dimension( A1 );
6
gap> Display( A1 );
GF(2)<x,y> / [ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
gap> unit1 := UniversalMorphismFromInitialObject( A1 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> Display( unit1 );
GF(2)<x,y> / [ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
 ^
 |
[  ]
 |
GF(2) / [  ]
gap> counit1 := Counit( A1, [ 1, 0 ] );
<A split epimorphism in CategoryOfFpAlgebras( GF(2) )>
gap> Display( counit1 );
GF(2) / [  ]
 ^
 |
[ 1*id(o), 0 ]
 |
GF(2)<x,y> / [ 1*x^2 + 1*x, 1*y^3 + 1*y, \
1*y⋅x + 1*x⋅y + 1*y ]
gap> o := A1.id_o;
1*id(o):(o) -≻ (o)
gap> x := A1.x;
1*x:(o) -≻ (o)
gap> y := A1.y;
1*y:(o) -≻ (o)
gap> comult1 := Comultiplication( A1,
>                    [ [ [ o, x ], [ x, o ], [ o, o ], [ y^2, y^2 ] ],
>                      [ [ o, y ], [ y, o + y^2 ] ] ] );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> q2 := FinQuiver( "q(o)[x:o->o,y:o->o,z:o->o,t:o->o]" );
FinQuiver( "q(o)[x:o-≻o,y:o-≻o,z:o-≻o,t:o-≻o]" )
gap> P2 := PathCategory( q2 );
PathCategory( FinQuiver( "q(o)[x:o-≻o,y:o-≻o,z:o-≻o,t:o-≻o]" ) )
gap> L2 := F2[P2];
GF(2)-LinearClosure( PathCategory( \
FinQuiver( "q(o)[x:o-≻o,y:o-≻o,z:o-≻o,t:o-≻o]" ) ) )
gap> Q2 := L2 / [ L2.x+L2.y+L2.t+L2.tt-L2.id_o, L2.xx-L2.x, L2.yy-L2.y, L2.xy, L2.yx,
>              L2.zz-L2.t-L2.tt, L2.zz-L2.ttt, L2.zt-L2.ttz, L2.zt+L2.ztt-L2.z,
>              L2.xt+L2.yt, L2.tx+L2.ty ];
GF(2)-LinearClosure( PathCategory( \
FinQuiver( "q(o)[x:o-≻o,y:o-≻o,z:o-≻o,t:o-≻o]" ) ) ) / \
[ 1*t^2 + 1*x + 1*y + 1*t + 1*id(o), 1*x^2 + 1*x, 1*y^2 + 1*y, ... ]
gap> A2 := Q2 / Alg_F2;
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Dimension( A2 );
6
gap> Display( A2 );
GF(2)<x,y,z,t> / [ 1*t^2 + 1*x + 1*y + 1*t + 1*id(o), 1*x^2 + 1*x, \
1*y^2 + 1*y, ... ]
gap> A12 := TensorProduct( A1, A2 );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Dimension( A12 );
36
gap> beta := Braiding( A1, A2 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> beta_i := BraidingInverse( A1, A2 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> alpha := AssociatorLeftToRight( A1, A2, A1 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> alpha_i := AssociatorRightToLeft( A1, A2, A1 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> A1xA2 := DirectProduct( A1, A2 );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Dimension( A1xA2 );
12
gap> diagram := [ A1, A2 ];
[ <An object in CategoryOfFpAlgebras( GF(2) )>,
  <An object in CategoryOfFpAlgebras( GF(2) )> ]
gap> pr1 := ProjectionInFactorOfDirectProduct( diagram, 1 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> pr2 := ProjectionInFactorOfDirectProduct( diagram, 2 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> u := UniversalMorphismIntoDirectProduct( [ pr1, pr2 ] );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> IsOne( u );
true
gap> v := TensorProduct( pr1, pr2 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> A1uA2 := Coproduct( A1, A2 );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> i1 := InjectionOfCofactorOfCoproduct( diagram, 1 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> i2 := InjectionOfCofactorOfCoproduct( diagram, 2 );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> w := UniversalMorphismFromCoproduct( [ i1, i2 ] );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> IsOne( w );
true
gap> cart_fact := LeftCartesianDistributivityFactoring( A1, [ A1, A2 ] );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> cocart_expand := LeftCocartesianCodistributivityExpanding( A1, [ A1, A2 ] );
<A morphism in CategoryOfFpAlgebras( GF(2) )>

The category of (finitely presented) algebras is not codistributive. An example was suggested by a referee in the paper the paper titled "Distributivity ot coproducts over products" by B. A. Davey and H. Werner, Algebra Universalis, 12 (1981) 387-394.

gap> LoadPackage( "Algebroids", false );
true
gap> F2 := HomalgRingOfIntegers( 2 );
GF(2)
gap> Alg_F2 := CategoryOfFpAlgebras( F2 );
CategoryOfFpAlgebras( GF(2) )
gap> F0 := InitialObject( Alg_F2 );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Display( F0 );
GF(2) / [  ]
gap> A := DirectProduct( F0, F0 );
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Dimension( A );
2
gap> B := F0;
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> C := F0;
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> eta := LeftCocartesianCodistributivityExpanding( A, [ B, C ] );
<A morphism in CategoryOfFpAlgebras( GF(2) )>
gap> IsWellDefined( eta );
true
gap> Dimension( Source( eta ) );
infinity
gap> Dimension( Target( eta ) );
4

10.3 Operations

10.3-1 Counit

10.3-2 Comultiplication

10.4 GAP categories

10.4-1 IsCategoryOfFinitelyPresentedAlgebras

The GAP category of categories of finitely presented associative unital algebras.

10.4-2 IsObjectInCategoryOfFpAlgebras

10.4-3 IsMorphismInCategoryOfFpAlgebras

The GAP category of morphisms of finitely presented associative unital algebras.

10 Category of finitely presented associative unital algebras

10.1 Attributes

10.1-1 CoefficientsRing