  
  [1X7 Relations[0X
  
  A finite presentation of a module is given by a finite set of generators and
  a  finite  set  of  relations  among  these  generators.  In [5Xhomalg[0m a set of
  relations  of a left/right module is given by a matrix [3Xrel[0m, the rows/columns
  of which are interpreted as relations among n generators, n being the number
  of columns/rows of the matrix [3Xrel[0m.
  
  The data structure of a module in [5Xhomalg[0m is designed to contain not only one
  but   several  sets  of  relations  (together  with  corresponding  sets  of
  generators (--> Chapter [14X8[0m)). The different sets of relations are linked with
  so-called transition matrices (--> Chapter [14X9[0m).
  
  The  relations  of  a [5Xhomalg[0m module are evaluated in a lazy way. This avoids
  unnecessary computations.
  
  
  [1X7.1 Relations: Categories and Representations[0X
  
  [1X7.1-1 IsHomalgRelations[0m
  
  [2X> IsHomalgRelations( [0X[3Xrel[0X[2X ) _________________________________________[0XCategory
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  The [5XGAP[0m category of [5Xhomalg[0m relations.
  
  [1X7.1-2 IsHomalgRelationsOfLeftModule[0m
  
  [2X> IsHomalgRelationsOfLeftModule( [0X[3Xrel[0X[2X ) _____________________________[0XCategory
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  The [5XGAP[0m category of [5Xhomalg[0m relations of a left module.
  
  (It is a subcategory of the [5XGAP[0m category [10XIsHomalgRelations[0m.)
  
  [1X7.1-3 IsHomalgRelationsOfRightModule[0m
  
  [2X> IsHomalgRelationsOfRightModule( [0X[3Xrel[0X[2X ) ____________________________[0XCategory
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  The [5XGAP[0m category of [5Xhomalg[0m relations of a right module.
  
  (It is a subcategory of the [5XGAP[0m category [10XIsHomalgRelations[0m.)
  
  [1X7.1-4 IsRelationsOfFinitelyPresentedModuleRep[0m
  
  [2X> IsRelationsOfFinitelyPresentedModuleRep( [0X[3Xrel[0X[2X ) _____________[0XRepresentation
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  The  [5XGAP[0m representation of a finite set of relations of a finitely presented
  [5Xhomalg[0m module.
  
  (It is a representation of the [5XGAP[0m category [2XIsHomalgRelations[0m ([14X7.1-1[0m))
  
  
  [1X7.2 Relations: Constructors[0X
  
  
  [1X7.3 Relations: Properties[0X
  
  [1X7.3-1 CanBeUsedToDecideZeroEffectively[0m
  
  [2X> CanBeUsedToDecideZeroEffectively( [0X[3Xrel[0X[2X ) __________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check  if  the  [5Xhomalg[0m  set  of  relations  [3Xrel[0m  can be used for normal form
  reductions.
  (no method installed)
  
  [1X7.3-2 IsInjectivePresentation[0m
  
  [2X> IsInjectivePresentation( [0X[3Xrel[0X[2X ) ___________________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check if the [5Xhomalg[0m set of relations [3Xrel[0m has zero syzygies.
  
  
  [1X7.4 Relations: Attributes[0X
  
  [1X7.4-1 FreeResolution[0m
  
  [2X> FreeResolution( [0X[3Xrel[0X[2X ) ___________________________________________[0Xattribute
  [6XReturns:[0X  a [5Xhomalg[0m complex
  
  The  computed  (part of a) free resolution of the module presented by [5Xhomalg[0m
  set of relations [3Xrel[0m.
  
  
  [1X7.5 Relations: Operations and Functions[0X
  
