  
  [1X5 [33X[0;0YRelations[133X[101X
  
  [33X[0;0YA finite presentation of a module is given by a finite set of generators and
  a  finite  set  of  relations  among  these  generators.  In [5Xhomalg[105X a set of
  relations  of a left/right module is given by a matrix [3Xrel[103X, the rows/columns
  of which are interpreted as relations among [22Xn[122X generators, [22Xn[122X being the number
  of columns/rows of the matrix [3Xrel[103X.[133X
  
  [33X[0;0YThe data structure of a module in [5Xhomalg[105X is designed to contain not only one
  but   several  sets  of  relations  (together  with  corresponding  sets  of
  generators (--> Chapter [14X6[114X)). The different sets of relations are linked with
  so-called transition matrices (--> Chapter [14X7[114X).[133X
  
  [33X[0;0YThe  relations  of  a [5Xhomalg[105X module are evaluated in a lazy way. This avoids
  unnecessary computations.[133X
  
  
  [1X5.1 [33X[0;0YRelations: Categories and Representations[133X[101X
  
  [1X5.1-1 IsHomalgRelations[101X
  
  [29X[2XIsHomalgRelations[102X( [3Xrel[103X ) [32X Category
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of [5Xhomalg[105X relations.[133X
  
  [1X5.1-2 IsHomalgRelationsOfLeftModule[101X
  
  [29X[2XIsHomalgRelationsOfLeftModule[102X( [3Xrel[103X ) [32X Category
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of [5Xhomalg[105X relations of a left module.[133X
  
  [33X[0;0Y(It is a subcategory of the [5XGAP[105X category [10XIsHomalgRelations[110X.)[133X
  
  [1X5.1-3 IsHomalgRelationsOfRightModule[101X
  
  [29X[2XIsHomalgRelationsOfRightModule[102X( [3Xrel[103X ) [32X Category
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of [5Xhomalg[105X relations of a right module.[133X
  
  [33X[0;0Y(It is a subcategory of the [5XGAP[105X category [10XIsHomalgRelations[110X.)[133X
  
  [1X5.1-4 IsRelationsOfFinitelyPresentedModuleRep[101X
  
  [29X[2XIsRelationsOfFinitelyPresentedModuleRep[102X( [3Xrel[103X ) [32X Representation
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe  [5XGAP[105X representation of a finite set of relations of a finitely presented
  [5Xhomalg[105X module.[133X
  
  [33X[0;0Y(It is a representation of the [5XGAP[105X category [2XIsHomalgRelations[102X ([14X5.1-1[114X))[133X
  
  
  [1X5.2 [33X[0;0YRelations: Constructors[133X[101X
  
  
  [1X5.3 [33X[0;0YRelations: Properties[133X[101X
  
  [1X5.3-1 CanBeUsedToDecideZeroEffectively[101X
  
  [29X[2XCanBeUsedToDecideZeroEffectively[102X( [3Xrel[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  the  [5Xhomalg[105X  set  of  relations  [3Xrel[103X  can be used for normal form
  reductions.[133X
  [33X[0;0Y(no method installed)[133X
  
  [1X5.3-2 IsInjectivePresentation[101X
  
  [29X[2XIsInjectivePresentation[102X( [3Xrel[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if the [5Xhomalg[105X set of relations [3Xrel[103X has zero syzygies.[133X
  
  
  [1X5.4 [33X[0;0YRelations: Attributes[133X[101X
  
  
  [1X5.5 [33X[0;0YRelations: Operations and Functions[133X[101X
  
