| ‣ FunctorStandardModuleLeft( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R. The output is a functor which takes a left presentation as input and computes its standard presentation.
| ‣ FunctorStandardModuleRight( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R. The output is a functor which takes a right presentation as input and computes its standard presentation.
| ‣ FunctorGetRidOfZeroGeneratorsLeft( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R. The output is a functor which takes a left presentation as input and gets rid of the zero generators.
| ‣ FunctorGetRidOfZeroGeneratorsRight( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R. The output is a functor which takes a right presentation as input and gets rid of the zero generators.
| ‣ FunctorLessGeneratorsLeft( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R. The output is functor which takes a left presentation as input and computes a presentation having less generators.
| ‣ FunctorLessGeneratorsRight( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R. The output is functor which takes a right presentation as input and computes a presentation having less generators.
| ‣ FunctorDualLeft( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R that has an involution function. The output is functor which takes a left presentation M as input and computes its Hom(M, R) as a left presentation.
| ‣ FunctorDualRight( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R that has an involution function. The output is functor which takes a right presentation M as input and computes its Hom(M, R) as a right presentation.
| ‣ FunctorDoubleDualLeft( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R that has an involution function. The output is functor which takes a left presentation M as input and computes its Hom( Hom(M, R), R ) as a left presentation.
| ‣ FunctorDoubleDualRight( R ) | ( attribute ) | 
Returns: a functor
The argument is a homalg ring R that has an involution function. The output is functor which takes a right presentation M as input and computes its Hom( Hom(M, R), R ) as a right presentation.
| ‣ IsLeftOrRightPresentationMorphism( object ) | ( filter ) | 
Returns: true or false
The GAP category of morphisms in the category of left or right presentations.
| ‣ IsLeftPresentationMorphism( object ) | ( filter ) | 
Returns: true or false
The GAP category of morphisms in the category of left presentations.
| ‣ IsRightPresentationMorphism( object ) | ( filter ) | 
Returns: true or false
The GAP category of morphisms in the category of right presentations.
| ‣ IsLeftOrRightPresentation( object ) | ( filter ) | 
Returns: true or false
The GAP category of objects in the category of left presentations or right presentations.
| ‣ IsLeftPresentation( object ) | ( filter ) | 
Returns: true or false
The GAP category of objects in the category of left presentations.
| ‣ IsRightPresentation( object ) | ( filter ) | 
Returns: true or false
The GAP category of objects in the category of right presentations.
| ‣ PresentationMorphism( A, M, B ) | ( operation ) | 
Returns: a morphism in \mathrm{Hom}(A,B)
The arguments are an object A, a homalg matrix M, and another object B. A and B shall either both be objects in the category of left presentations or both be objects in the category of right presentations. The output is a morphism A \rightarrow B in the the category of left or right presentations whose underlying matrix is given by M.
| ‣ AsMorphismBetweenFreeLeftPresentations( m ) | ( attribute ) | 
Returns: a morphism in \mathrm{Hom}(F^r,F^c)
The argument is a homalg matrix m. The output is a morphism F^r \rightarrow F^c in the the category of left presentations whose underlying matrix is given by m, where F^r and F^c are free left presentations of ranks given by the number of rows and columns of m.
| ‣ AsMorphismBetweenFreeRightPresentations( m ) | ( attribute ) | 
Returns: a morphism in \mathrm{Hom}(F^c,F^r)
The argument is a homalg matrix m. The output is a morphism F^c \rightarrow F^r in the the category of right presentations whose underlying matrix is given by m, where F^r and F^c are free right presentations of ranks given by the number of rows and columns of m.
| ‣ AsLeftPresentation( M ) | ( operation ) | 
Returns: an object
The argument is a homalg matrix M over a ring R. The output is an object in the category of left presentations over R. This object has M as its underlying matrix.
| ‣ AsRightPresentation( M ) | ( operation ) | 
Returns: an object
The argument is a homalg matrix M over a ring R. The output is an object in the category of right presentations over R. This object has M as its underlying matrix.
| ‣ AsLeftOrRightPresentation( M, l ) | ( function ) | 
Returns: an object
The arguments are a homalg matrix M and a boolean l. If l is true, the output is an object in the category of left presentations. If l is false, the output is an object in the category of right presentations. In both cases, the underlying matrix of the result is M.
| ‣ FreeLeftPresentation( r, R ) | ( operation ) | 
Returns: an object
The arguments are a non-negative integer r and a homalg ring R. The output is an object in the category of left presentations over R. It is represented by the 0 \times r matrix and thus it is free of rank r.
| ‣ FreeRightPresentation( r, R ) | ( operation ) | 
Returns: an object
The arguments are a non-negative integer r and a homalg ring R. The output is an object in the category of right presentations over R. It is represented by the r \times 0 matrix and thus it is free of rank r.
| ‣ UnderlyingMatrix( A ) | ( attribute ) | 
Returns: a homalg matrix
The argument is an object A in the category of left or right presentations over a homalg ring R. The output is the underlying matrix which presents A.
| ‣ UnderlyingHomalgRing( A ) | ( attribute ) | 
Returns: a homalg ring
The argument is an object A in the category of left or right presentations over a homalg ring R. The output is R.
| ‣ Annihilator( A ) | ( attribute ) | 
Returns: a morphism in \mathrm{Hom}(I, F)
The argument is an object A in the category of left or right presentations. The output is the embedding of the annihilator I of A into the free module F of rank 1. In particular, the annihilator itself is seen as a left or right presentation.
| ‣ LeftPresentations( R ) | ( attribute ) | 
Returns: a category
The argument is a homalg ring R. The output is the category of free left presentations over R.
| ‣ RightPresentations( R ) | ( attribute ) | 
Returns: a category
The argument is a homalg ring R. The output is the category of free right presentations over R.
| ‣ UnderlyingHomalgRing( R ) | ( attribute ) | 
Returns: a homalg ring
The argument is a morphism \alpha in the category of left or right presentations over a homalg ring R. The output is R.
| ‣ UnderlyingMatrix( alpha ) | ( attribute ) | 
Returns: a homalg matrix
The argument is a morphism \alpha in the category of left or right presentations. The output is its underlying homalg matrix.
| ‣ StandardGeneratorMorphism( A, i ) | ( operation ) | 
Returns: a morphism in \mathrm{Hom}(F, A)
The argument is an object A in the category of left or right presentations over a homalg ring R with underlying matrix M and an integer i. The output is a morphism F \rightarrow A given by the i-th row or column of M, where F is a free left or right presentation of rank 1.
| ‣ CoverByFreeModule( A ) | ( attribute ) | 
Returns: a morphism in \mathrm{Hom}(F,A)
The argument is an object A in the category of left or right presentations. The output is a morphism from a free module F to A, which maps the standard generators of the free module to the generators of A.
| ‣ NaturalIsomorphismFromIdentityToStandardModuleLeft( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{StandardModuleLeft}
The argument is a homalg ring R. The output is the natural isomorphism from the identity functor to the left standard module functor.
| ‣ NaturalIsomorphismFromIdentityToStandardModuleRight( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{StandardModuleRight}
The argument is a homalg ring R. The output is the natural isomorphism from the identity functor to the right standard module functor.
| ‣ NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsLeft}
The argument is a homalg ring R. The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of left modules.
| ‣ NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsRight}
The argument is a homalg ring R. The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of right modules.
| ‣ NaturalIsomorphismFromIdentityToLessGeneratorsLeft( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{LessGeneratorsLeft}
The argument is a homalg ring R. The output is the natural morphism from the identity functor to the left less generators functor.
| ‣ NaturalIsomorphismFromIdentityToLessGeneratorsRight( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{LessGeneratorsRight}
The argument is a homalg ring R. The output is the natural morphism from the identity functor to the right less generator functor.
| ‣ NaturalTransformationFromIdentityToDoubleDualLeft( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualLeft}
The argument is a homalg ring R. The output is the natural morphism from the identity functor to the double dual functor in left Presentations category.
| ‣ NaturalTransformationFromIdentityToDoubleDualRight( R ) | ( attribute ) | 
Returns: a natural transformation \mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualRight}
The argument is a homalg ring R. The output is the natural morphism from the identity functor to the double dual functor in right Presentations category.
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