  
  [1XF [33X[0;0YOverview of the [5XMatricesForHomalg[105X[101X[1X Package Source Code[133X[101X
  
  
  [1XF.1 [33X[0;0YRings, Ring Maps, Matrices, Ring Relations[133X[101X
  
        Filename [11X.gd[111X/[11X.gi[111X    │ Content                                     
        ────────────────────┼──────────────────────────────────────────  
        [11Xhomalg[111X              │ definitions of the basic [5XGAP4[105X categories    
                            │ and some tool functions (e.g. [10XhomalgMode[110X)   
                            │                                             
        [11XhomalgTable[111X         │ dictionaries between [5XMatricesForHomalg[105X      
                            │ and the computing engines                   
        [11XHomalgRing[111X          │ internal and external rings                 
                            │                                             
        [11XHomalgRingMap[111X       │ ring maps                                   
                            │                                             
        [11XHomalgMatrix[111X        │ internal and external matrices              
                            │                                             
        [11XHomalgRingRelations[111X │ a set of ring relations                     
  
       [1XTable:[101X [13XThe [5XMatricesForHomalg[105X package files[113X
  
  
  
  [1XF.2 [33X[0;0YThe Low Level Algorithms[133X[101X
  
  [33X[0;0YIn the following CAS or CASystem mean computer algebra systems.[133X
  
        Filename [11X.gd[111X/[11X.gi[111X │ Content                                                 
        ─────────────────┼──────────────────────────────────────────────────────  
        [11XTools[111X            │ the elementary matrix operations that can be            
                         │ overwritten using the homalgTable                       
                         │ (and hence delegable even to other CASystems)           
                         │                                                         
        [11XService[111X          │ the three operations: basis, reduction, and syzygies;   
                         │ they can also be overwritten using the homalgTable      
                         │ (and hence delegable even to other CASystems)           
                         │                                                         
        [11XBasic[111X            │ higher level operations for matrices                    
                         │ (cannot be overwritten using the homalgTable)           
  
       [1XTable:[101X [13XThe [5XMatricesForHomalg[105X package files (continued)[113X
  
  
  
  [1XF.3 [33X[0;0YLogical Implications for [5XMatricesForHomalg[105X[101X[1X Objects[133X[101X
  
        Filename [11X.gd[111X/[11X.gi[111X │ Content                                         
        ─────────────────┼──────────────────────────────────────────────  
        [11XLIRNG[111X            │ logical implications for rings                  
                         │                                                 
        [11XLIMAP[111X            │ logical implications for ring maps              
                         │                                                 
        [11XLIMAT[111X            │ logical implications for matrices               
                         │                                                 
        [11XCOLEM[111X            │ clever operations for lazy evaluated matrices   
                         │                                                 
  
       [1XTable:[101X [13XThe [5XMatricesForHomalg[105X package files (continued)[113X
  
  
  
  [1XF.4 [33X[0;0YThe subpackage [5XResidueClassRingForHomalg[105X[101X[1X[133X[101X
  
        Filename [11X.gd[111X/[11X.gi[111X          │ Content                                                
        ──────────────────────────┼─────────────────────────────────────────────────────  
        [11XResidueClassRingForHomalg[111X │ some global variables                                  
                                  │                                                        
        [11XResidueClassRing[111X          │ residue class rings, their elements, and matrices,     
                                  │ together with their constructors and operations        
                                  │                                                        
        [11XResidueClassRingTools[111X     │ the elementary matrix operations for matrices          
                                  │ over residue class rings                               
                                  │                                                        
        [11XResidueClassRingBasic[111X     │ the three operations: basis, reduction, and syzygies   
                                  │ for matrices over residue class rings                  
                                  │                                                        
  
       [1XTable:[101X [13XThe [5XMatricesForHomalg[105X package files (continued)[113X
  
  
  
  [1XF.5 [33X[0;0YThe homalgTable for [5XGAP4[105X[101X[1X built-in rings[133X[101X
  
  [33X[0;0YFor  the  purposes of [5Xhomalg[105X, the ring of integers is, at least up till now,
  the  only  ring  which  is  properly  supported  in  [5XGAP4[105X. The [5XGAP4[105X built-in
  cababilities  for  polynomial rings (also univariate) and group rings do not
  statisfy  the minimum requirements of [5Xhomalg[105X. The [5XGAP4[105X package [5XGauss[105X enables
  [5XGAP[105X to fullfil the [5Xhomalg[105X requirements for prime fields, and [22Xℤ / p^n[122X.[133X
  
        Filename .gi │ Content                                    
        ─────────────┼─────────────────────────────────────────  
        [11XIntegers[111X     │ the homalgTable for the ring of integers   
  
       [1XTable:[101X [13XThe [5XMatricesForHomalg[105X package files (continued)[113X
  
  
