chap6.txt

  
  6 Generators
  
  To  present  a  left/right  module  it  suffices  to  take  a matrix rel and
  interpret its rows/columns as relations among n abstract generators, where n
  is  the  number  of columns/rows of rel. Only that these abstract generators
  are useless when it comes to specific modules like modules of homomorphisms,
  where one expects the generators to be maps between modules. For this reason
  a  presentation  of  a module in homalg is not merely a matrix of relations,
  but together with a set of generators.
  
  In  homalg  a  set of generators of a left/right module is given by a matrix
  gen with rows/columns being interpreted as the generators.
  
  The data structure of a module in homalg is designed to contain not only one
  but  several  sets of generators (together with their sets of relations (-->
  Chapter  5)).  The  different  sets  of generators are linked with so-called
  transition matrices (--> Chapter 7).
  
  
  6.1 Generators: Categories and Representations
  
  6.1-1 IsHomalgGenerators
  
  IsHomalgGenerators( rel )  Category
  Returns:  true or false
  
  The GAP category of homalg generators.
  
  6.1-2 IsHomalgGeneratorsOfLeftModule
  
  IsHomalgGeneratorsOfLeftModule( rel )  Category
  Returns:  true or false
  
  The GAP category of homalg generators of a left module.
  
  (It is a subcategory of the GAP category IsHomalgGenerators.)
  
  6.1-3 IsHomalgGeneratorsOfRightModule
  
  IsHomalgGeneratorsOfRightModule( rel )  Category
  Returns:  true or false
  
  The GAP category of homalg generators of a right module.
  
  (It is a subcategory of the GAP category IsHomalgGenerators.)
  
  6.1-4 IsGeneratorsOfModuleRep
  
  IsGeneratorsOfModuleRep( rel )  Representation
  Returns:  true or false
  
  The GAP representation of a finite set of generators of a homalg module.
  
  (It is a representation of the GAP category IsHomalgGenerators (6.1-1))
  
    Code  
    DeclareRepresentation( "IsGeneratorsOfModuleRep",
            IsHomalgGenerators,
            [ "generators" ] );
  
  
  6.1-5 IsGeneratorsOfFinitelyGeneratedModuleRep
  
  IsGeneratorsOfFinitelyGeneratedModuleRep( rel )  Representation
  Returns:  true or false
  
  The GAP representation of a finite set of generators of a finitely generated
  homalg module.
  
  (It  is  a  representation of the GAP representation IsGeneratorsOfModuleRep
  (6.1-4))
  
    Code  
    DeclareRepresentation( "IsGeneratorsOfFinitelyGeneratedModuleRep",
            IsGeneratorsOfModuleRep,
            [ "generators", "relations_of_hullmodule" ] );
  
  
  
  6.2 Generators: Constructors
  
  
  6.3 Generators: Properties
  
  6.3-1 IsReduced
  
  IsReduced( gen )  property
  Returns:  true or false
  
  Check if the homalg set of generators gen is marked reduced.
  (no method installed)
  
  
  6.4 Generators: Attributes
  
  6.4-1 ProcedureToReadjustGenerators
  
  ProcedureToReadjustGenerators( gen )  attribute
  Returns:  a function
  
  A  function that takes the rows/columns of gen and returns an object (e.g. a
  matrix)  that  can  be  interpreted  as  a  generator (this is important for
  modules of homomorphisms).
  
  
  6.5 Generators: Operations and Functions