9 Category of graded rows and category of graded columns

9.4 Tools to simplify code

9.4-1 DeduceMapFromMatrixAndRangeForGradedRows

9.4-2 DeduceSomeMapFromMatrixAndRangeForGradedRows

9.4-3 DeduceMapFromMatrixAndSourceForGradedRows

9.4-4 DeduceSomeMapFromMatrixAndSourceForGradedRows

9.4-5 DeduceMapFromMatrixAndRangeForGradedCols

9.4-6 DeduceSomeMapFromMatrixAndRangeForGradedCols

9.4-7 DeduceMapFromMatrixAndSourceForGradedCols

9.4-8 DeduceSomeMapFromMatrixAndSourceForGradedCols

9.4-9 UnzipDegreeList

9.4-1 DeduceMapFromMatrixAndRangeForGradedRows

9.4-2 DeduceSomeMapFromMatrixAndRangeForGradedRows

9.4-3 DeduceMapFromMatrixAndSourceForGradedRows

9.4-4 DeduceSomeMapFromMatrixAndSourceForGradedRows

9.4-5 DeduceMapFromMatrixAndRangeForGradedCols

9.4-6 DeduceSomeMapFromMatrixAndRangeForGradedCols

9.4-7 DeduceMapFromMatrixAndSourceForGradedCols

9.4-8 DeduceSomeMapFromMatrixAndSourceForGradedCols

9.4-9 UnzipDegreeList

`‣ CategoryOfGradedColumns` ( R ) | ( attribute ) |

Returns: a category

The argument is a homalg graded ring \(R\). The output is the category of graded columns over \(R\).

`‣ CategoryOfGradedRows` ( R ) | ( attribute ) |

Returns: a category

The argument is a homalg graded ring \(R\). The output is the category of graded rows over \(R\).

`‣ GradedRow` ( degree_list, R ) | ( operation ) |

Returns: an object

The arguments are a list of degrees and a homalg graded ring \(R\). The list of degrees must be of the form [ [ \(d_1\), \(n_1\) ], [ \(d_2\), \(n_2\) ], ... ] where \(d_i\) are degrees, i.e. elements in the degree group of \(R\) and the \(n_i\) are non-negative integers. Currently there are two formats that are supported to enter the degrees. Either one can enter them as lists of integers, say \(d_1 = [ 1,1,0,2 ]\), or they can be entered as Homalg_Module_Elements of the degree group of \(R\). In either case, the result is the graded row associated to the degrees \(d_i\) and their multiplicities \(n_i\).

`‣ GradedRow` ( degree_list, R ) | ( operation ) |

Returns: an object

As 'GradedRow', but the boolean (= third argument) allows to switch off checks on the input data. If this boolean is set to true, then the input checks are performed and otherwise they are not. Calling this constructor with 'false' is therefore suited for high performance applications.

`‣ GradedColumn` ( degree_list, R ) | ( operation ) |

Returns: an object

The arguments are a list of degrees and a homalg graded ring \(R\). The list of degrees must be of the form [ [ \(d_1\), \(n_1\) ], [ \(d_2\), \(n_2\) ], ... ] where \(d_i\) are degrees, i.e. elements in the degree group of \(R\) and the \(n_i\) are non-negative integers. Currently there are two formats that are supported to enter the degrees. Either one can enter them as lists of integers, say \(d_1 = [ 1,1,0,2 ]\), or they can be entered as Homalg_Module_Elements of the degree group of \(R\). In either case, the result is the graded column associated to the degrees \(d_i\) and their multiplicities \(n_i\).

`‣ GradedColumn` ( degree_list, R ) | ( operation ) |

Returns: an object

As 'GradedColumn', but the boolean (= third argument) allows to switch off checks on the input data. If this boolean is set to true, then the input checks are performed and otherwise they are not. Calling this constructor with 'false' is therefore suited for high performance applications.

`‣ GradedRowOrColumnMorphism` ( S, M, T ) | ( operation ) |

Returns: a morphism in \(\mathrm{Hom}(S,T)\)

The arguments are an object \(S\) in the category of graded rows or columns over a homalg graded ring \(R\), a homalg matrix \(M\) over \(R\) and another graded row or column \(T\) over \(R\). The output is the morphism \(S \rightarrow T\) in the category of graded rows and columns over \(R\), whose underlying matrix is given by \(M\).

`‣ GradedRowOrColumnMorphism` ( S, M, T ) | ( operation ) |

Returns: a morphism in \(\mathrm{Hom}(S,T)\)

As 'GradedRowOrColumnMorphism', but carries a fourth input parameter. If this boolean is set to false, then no checks on the input a performed. That option is therefore better suited for high performance applications.

`‣ UnderlyingHomalgGradedRing` ( A ) | ( attribute ) |

Returns: a homalg graded ring

The argument is a graded row or column \(A\) over a homalg graded ring \(R\). The output is then the graded ring \(R\).

`‣ DegreeList` ( A ) | ( attribute ) |

Returns: a list

The argument is a graded row or column \(A\) over a homalg graded ring \(R\). The output is the degree_list of this object. To handle degree_lists most easily, degree_lists are redcued whenever an object is added to the category. E.g. the input degree_list [ [ \(d_1\), 1 ], [ \(d_1\), 1 ] ] will be turned into [ [ \(d_1\), 2 ] ].

`‣ RankOfObject` ( A ) | ( attribute ) |

Returns: an integer

The argument is a graded row or column over a homalg graded ring \(R\). The output is the rank of this module.

`‣ UnderlyingHomalgGradedRing` ( alpha ) | ( attribute ) |

Returns: a homalg graded ring

The argument is a morphism \(\alpha\) in the category of graded rows or columns over a homalg graded ring \(R\). The output is the homalg graded ring \(R\).

`‣ UnderlyingHomalgMatrix` ( alpha ) | ( attribute ) |

Returns: a matrix over a homalg graded ring

The argument is a morphism \(\alpha\) in the category of graded rows or columns over a homalg graded ring \(R\). The output is the underlying homalg matrix over \(R\).

`‣ IsGradedRowOrColumn` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of graded rows and columns over a graded ring \(R\).

`‣ IsGradedRow` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of graded rows over a graded ring \(R\).

`‣ IsGradedColumn` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of graded columns over a graded ring \(R\).

`‣ IsGradedRowOrColumnMorphism` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of morphisms of graded rows and columns over a graded ring \(R\).

`‣ IsGradedRowMorphism` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of morphisms of graded rows over a graded ring \(R\).

`‣ IsGradedColumnMorphism` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of morphisms of graded columns over a graded ring \(R\).

`‣ DeduceMapFromMatrixAndRangeForGradedRows` ( m, R ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded row `R`. We then consider the module map induced from `m` with range `R`. This operation then deduces the source of this map and returns the map in the category of graded rows if the degrees of the source are uniquely determined.

`‣ DeduceSomeMapFromMatrixAndRangeForGradedRows` ( m, R ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded row `R`. This operation deduces the source of some map with matrix `m` and range `R` and returns the map in the category of graded rows.

`‣ DeduceMapFromMatrixAndSourceForGradedRows` ( m, S ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded row `S`. We then consider the module map induced from `m` with source `S`. This operation then deduces the range of this map and returns the map in the category of graded rows if the degrees of the range are uniquely determined.

`‣ DeduceSomeMapFromMatrixAndSourceForGradedRows` ( m, S ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded row `S`. This operation deduces the range of some map with matrix `m` and source `S` and returns the map in the category of graded rows.

`‣ DeduceMapFromMatrixAndRangeForGradedCols` ( m, R ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded column `R`. We then consider the module map induced from `m` with range `R`. This operation then deduces the source of this map and returns the map in the category of graded columns if the degrees of the source are uniquely determined.

`‣ DeduceSomeMapFromMatrixAndRangeForGradedCols` ( m, R ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded column `R`. This operation deduces the source of some map with matrix `m` and range `R` and returns the map in the category of graded columns.

`‣ DeduceMapFromMatrixAndSourceForGradedCols` ( m, S ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded column `S`. We then consider the module map induced from `m` with source `S`. This operation then deduces the range of this map and returns the map in the category of graded columns if the degrees of the range are uniquely determined.

`‣ DeduceSomeMapFromMatrixAndSourceForGradedCols` ( m, S ) | ( operation ) |

Returns: a morphism

The argument is a homalg_matrix `m` and a graded column `S`. This operation deduces the range of some map with matrix `m` and source `S` and returns the map in the category of graded columns.

`‣ UnzipDegreeList` ( S ) | ( operation ) |

Returns: a list

Given a graded row or column `S`, the degrees are stored in compact form. For example, the degrees [ 1, 1, 1, 1 ] #! is stored internally as [ 1, 4 ]. The second argument is thus the multipicity with which three degree 1 appears. Still, it can be useful at times to also go in the opposite direction, i.e. to take the compact form [ #! 1, 4 ] and turn it into [ 1, 1, 1, 1 ]. This is performed by this operation and the obtained extended degree #! list is returned.

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