chap5.txt

  
  5 Relations
  
  A finite presentation of a module is given by a finite set of generators and
  a  finite  set  of  relations  among  these  generators.  In homalg a set of
  relations  of a left/right module is given by a matrix rel, the rows/columns
  of which are interpreted as relations among n generators, n being the number
  of columns/rows of the matrix rel.
  
  The data structure of a module in homalg is designed to contain not only one
  but   several  sets  of  relations  (together  with  corresponding  sets  of
  generators (--> Chapter 6)). The different sets of relations are linked with
  so-called transition matrices (--> Chapter 7).
  
  The  relations  of  a homalg module are evaluated in a lazy way. This avoids
  unnecessary computations.
  
  
  5.1 Relations: Categories and Representations
  
  5.1-1 IsHomalgRelations
  
  IsHomalgRelations( rel )  Category
  Returns:  true or false
  
  The GAP category of homalg relations.
  
  5.1-2 IsHomalgRelationsOfLeftModule
  
  IsHomalgRelationsOfLeftModule( rel )  Category
  Returns:  true or false
  
  The GAP category of homalg relations of a left module.
  
  (It is a subcategory of the GAP category IsHomalgRelations.)
  
  5.1-3 IsHomalgRelationsOfRightModule
  
  IsHomalgRelationsOfRightModule( rel )  Category
  Returns:  true or false
  
  The GAP category of homalg relations of a right module.
  
  (It is a subcategory of the GAP category IsHomalgRelations.)
  
  5.1-4 IsRelationsOfFinitelyPresentedModuleRep
  
  IsRelationsOfFinitelyPresentedModuleRep( rel )  Representation
  Returns:  true or false
  
  The  GAP representation of a finite set of relations of a finitely presented
  homalg module.
  
  (It is a representation of the GAP category IsHomalgRelations (5.1-1))
  
  
  5.2 Relations: Constructors
  
  
  5.3 Relations: Properties
  
  5.3-1 CanBeUsedToDecideZeroEffectively
  
  CanBeUsedToDecideZeroEffectively( rel )  property
  Returns:  true or false
  
  Check  if  the  homalg  set  of  relations  rel  can be used for normal form
  reductions.
  (no method installed)
  
  5.3-2 IsInjectivePresentation
  
  IsInjectivePresentation( rel )  property
  Returns:  true or false
  
  Check if the homalg set of relations rel has zero syzygies.
  
  
  5.4 Relations: Attributes
  
  
  5.5 Relations: Operations and Functions