File: mapping.gd

package info (click to toggle)
gap 4r8p6-2
  • links: PTS
  • area: main
  • in suites: stretch
  • size: 33,476 kB
  • ctags: 7,663
  • sloc: ansic: 108,841; xml: 47,807; sh: 3,628; perl: 2,342; makefile: 796; asm: 62; awk: 6
file content (1484 lines) | stat: -rw-r--r-- 54,579 bytes parent folder | download | duplicates (2)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
#############################################################################
##
#W  mapping.gd                  GAP library                     Thomas Breuer
#W                                                         & Martin Schönert
#W                                                             & Frank Celler
##
##
#Y  Copyright (C)  1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
#Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
##
##  This file declares the operations for general mappings.
##

#############################################################################
##
##  <#GAPDoc Label="[1]{mapping}">
##  A <E>general mapping</E> <M>F</M> in &GAP; is described by
##  its source <M>S</M>, its range <M>R</M>, and a subset <M>Rel</M> of the
##  direct product <M>S \times R</M>,
##  which is called the underlying relation of <M>F</M>.
##  <M>S</M>, <M>R</M>, and <M>Rel</M> are generalized domains
##  (see <Ref Chap="Domains"/>).
##  The corresponding attributes for general mappings are
##  <Ref Func="Source"/>, <Ref Func="Range" Label="of a general mapping"/>,
##  and <Ref Func="UnderlyingRelation"/>.
##  <!-- what about the family predicates if the source/range is not a -->
##  <!-- collection? -->
##  <P/>
##  Note that general mappings themselves are <E>not</E> domains.
##  One reason for this is that two general mappings with same underlying
##  relation are regarded as equal only if also the sources are equal and
##  the ranges are equal.
##  Other, more technical, reasons are that general mappings and domains
##  have different basic operations, and that general mappings are
##  arithmetic objects
##  (see&nbsp;<Ref Sect="Arithmetic Operations for General Mappings"/>);
##  both should not apply to domains.
##  <P/>
##  Each element of an underlying relation of a general mapping lies in the
##  category of direct product elements
##  (see&nbsp;<Ref Func="IsDirectProductElement"/>).
##  <P/>
##  For each <M>s \in S</M>, the set <M>\{ r \in R | (s,r) \in Rel \}</M>
##  is called the set of <E>images</E> of <M>s</M>.
##  Analogously, for <M>r \in R</M>,
##  the set <M>\{ s \in S | (s,r) \in Rel \}</M>
##  is called the set of <E>preimages</E> of <M>r</M>.
##  <P/>
##  The <E>ordering</E> of general mappings via <C>&lt;</C> is defined
##  by the ordering of source, range, and underlying relation.
##  Specifically, if the source and range domains of <A>map1</A> and
##  <A>map2</A> are the same, then one considers  the union of the preimages
##  of <A>map1</A> and <A>map2</A> as a strictly ordered set.
##  The smaller of <A>map1</A> and <A>map2</A> is the one whose image is
##  smaller on the  first point of this sequence where they differ.
##  <#/GAPDoc>
##
##  <#GAPDoc Label="[2]{mapping}">
##  <Ref Func="Source"/> and <Ref Func="Range" Label="of a general mapping"/> 
##  are basic operations for general mappings.
##  <Ref Func="UnderlyingRelation"/> is secondary, its default method sets up
##  a domain that delegates tasks to the general mapping.
##  (Note that this allows one to handle also infinite relations by generic
##  methods if source or range of the general mapping is finite.)
##  <P/>
##  The distinction between basic operations and secondary operations for
##  general mappings may be a little bit complicated.
##  Namely, each general mapping must be in one of the two categories
##  <Ref Func="IsNonSPGeneralMapping"/>, <Ref Func="IsSPGeneralMapping"/>.
##  (The category <Ref Func="IsGeneralMapping"/> is defined as the disjoint
##  union of these two.)
##  <P/>
##  For general mappings of the first category, <Ref Func="ImagesElm"/> and
##  <Ref Func="PreImagesElm"/> are basic operations.
##  (Note that in principle it is possible to delegate
##  from <Ref Func="PreImagesElm"/> to <Ref Func="ImagesElm"/>.)
##  Methods for the secondary operations <Ref Func="ImageElm"/>,
##  <Ref Func="PreImageElm"/>, <Ref Func="ImagesSet"/>,
##  <Ref Func="PreImagesSet"/>, <Ref Func="ImagesRepresentative"/>,
##  and <Ref Func="PreImagesRepresentative"/> may use
##  <Ref Func="ImagesElm"/> and <Ref Func="PreImagesElm"/>, respectively,
##  and methods for <Ref Func="ImagesElm"/>, <Ref Func="PreImagesElm"/>
##  must <E>not</E> call the secondary operations.
##  In particular, there are no generic methods for
##  <Ref Func="ImagesElm"/> and <Ref Func="PreImagesElm"/>.
##  <P/>
##  Methods for <Ref Func="ImagesSet"/> and <Ref Func="PreImagesSet"/> must
##  <E>not</E> use <Ref Func="PreImagesRange"/> and
##  <Ref Func="ImagesSource"/>, e.g.,
##  compute the intersection of the set in question with the preimage of the
##  range resp. the image of the source.
##  <P/>
##  For general mappings of the second category (which means structure
##  preserving general mappings), the situation is different.
##  The set of preimages under a group homomorphism, for example, is either
##  empty or can be described as a coset of the (multiplicative) kernel.
##  So it is reasonable to have <Ref Func="ImagesRepresentative"/>,
##  <Ref Func="PreImagesRepresentative"/>,
##  <Ref Func="KernelOfMultiplicativeGeneralMapping"/>, and
##  <Ref Func="CoKernelOfMultiplicativeGeneralMapping"/> as basic operations
##  here, and to make <Ref Func="ImagesElm"/> and <Ref Func="PreImagesElm"/>
##  secondary operations that may delegate to these.
##  <P/>
##  In order to avoid infinite recursions,
##  we must distinguish between the two different types of mappings.
##  <P/>
##  (Note that the basic domain operations such as <Ref Func="AsList"/>
##  for the underlying relation of a general mapping may use either
##  <Ref Func="ImagesElm"/> or <Ref Func="ImagesRepresentative"/> and the
##  appropriate cokernel.
##  Conversely, if <Ref Func="AsList"/> for the underlying relation is known
##  then <Ref Func="ImagesElm"/> resp. <Ref Func="ImagesRepresentative"/>
##  may delegate to it,
##  the general mapping gets the property
##  <Ref Func="IsConstantTimeAccessGeneralMapping"/> for this;
##  note that this is not allowed if only an enumerator of the underlying
##  relation is known.)
##  <P/>
##  Secondary operations are
##  <Ref Func="IsInjective"/>, <Ref Func="IsSingleValued"/>,
##  <Ref Func="IsSurjective"/>, <Ref Func="IsTotal"/>;
##  they may use the basic operations, and must not be used by them.
##  <#/GAPDoc>
##
##  <#GAPDoc Label="[3]{mapping}">
##  General mappings are arithmetic objects.
##  One can form groups and vector spaces of general mappings provided
##  that they are invertible or can be added and admit scalar multiplication,
##  respectively.
##  <P/>
##  For two general mappings with same source, range, preimage, and image,
##  the <E>sum</E> is defined pointwise, i.e.,
##  the images of a point under the sum is the set of all sums with
##  first summand in the images of the first general mapping and
##  second summand in the images of the second general mapping.
##  <P/>
##  <E>Scalar multiplication</E> of general mappings is defined likewise.
##  <P/>
##  The <E>product</E> of two general mappings is defined as the composition.
##  This multiplication is always associative.
##  In addition to the composition via <C>*</C>,
##  general mappings can be composed &ndash;in reversed order&ndash;
##  via <Ref Func="CompositionMapping"/>.
##  <P/>
##  General mappings are in the category of multiplicative elements with
##  inverses.
##  Similar to matrices, not every general mapping has an inverse or an
##  identity, and we define the behaviour of <Ref Func="One"/> and
##  <Ref Func="Inverse"/> for general mappings as follows.
##  <Ref Func="One"/> returns <K>fail</K> when called for a general mapping
##  whose source and range differ,
##  otherwise <Ref Func="One"/> returns the identity mapping of the source.
##  (Note that the source may differ from the preimage).
##  <Ref Func="Inverse"/> returns <K>fail</K> when called for a non-bijective
##  general mapping or for a general mapping whose source and range differ;
##  otherwise <Ref Func="Inverse"/> returns the inverse mapping.
##  <P/>
##  Besides the usual inverse of multiplicative elements, which means that
##  <C>Inverse( <A>g</A> ) * <A>g</A> = <A>g</A> * Inverse( <A>g</A> )
##  = One( <A>g</A> )</C>,
##  for general mappings we have the attribute
##  <Ref Func="InverseGeneralMapping"/>.
##  If <A>F</A> is a general mapping with source <M>S</M>, range <M>R</M>,
##  and underlying relation <M>Rel</M> then
##  <C>InverseGeneralMapping( <A>F</A> )</C> has source <M>R</M>,
##  range <M>S</M>,
##  and underlying relation <M>\{ (r,s) \mid (s,r) \in Rel \}</M>.
##  For a general mapping that has an inverse in the usual sense,
##  i.e., for a bijection of the source, of course both concepts coincide.
##  <P/>
##  <Ref Func="Inverse"/> may delegate to
##  <Ref Func="InverseGeneralMapping"/>.
##  <Ref Func="InverseGeneralMapping"/> must not delegate to
##  <Ref Func="Inverse"/>,
##  but a known value of <Ref Func="Inverse"/> may be fetched.
##  So methods to compute the inverse of a general mapping should be
##  installed for <Ref Func="InverseGeneralMapping"/>.
##  <P/>
##  (Note that in many respects, general mappings behave similar to matrices,
##  for example one can define left and right identities and inverses, which
##  do not fit into the current concepts of &GAP;.)
##  <#/GAPDoc>
##
##  <#GAPDoc Label="[4]{mapping}">
##  Methods for the operations <Ref Func="ImagesElm"/>,
##  <Ref Func="ImagesRepresentative"/>,
##  <Ref Func="ImagesSet"/>, <Ref Func="ImageElm"/>,
##  <Ref Func="PreImagesElm"/>,
##  <Ref Func="PreImagesRepresentative"/>, <Ref Func="PreImagesSet"/>,
##  and <Ref Func="PreImageElm"/> take two arguments, a general mapping
##  <A>map</A> and an element or collection of elements <A>elm</A>.
##  These methods must <E>not</E> check whether <A>elm</A> lies in the source
##  or the range of <A>map</A>.
##  In the case that <A>elm</A> does not, <K>fail</K> may be returned as well
##  as any other &GAP; object, and even an error message is allowed.
##  Checks of the arguments are done only by the functions
##  <Ref Func="Image" Label="set of images of the source of a general mapping"/>,
##  <Ref Func="Images" Label="set of images of the source of a general mapping"/>,
##  <Ref Func="PreImage" Label="set of preimages of the range of a general mapping"/>,
##  and <Ref Func="PreImages" Label="set of preimages of the range of a general mapping"/>,
##  which then delegate to the operations listed above.
##  <#/GAPDoc>
##


#############################################################################
##
#C  IsGeneralMapping( <map> )
##
##  <#GAPDoc Label="IsGeneralMapping">
##  <ManSection>
##  <Filt Name="IsGeneralMapping" Arg='map' Type='Category'/>
##
##  <Description>
##  Each general mapping lies in the category <Ref Func="IsGeneralMapping"/>.
##  It implies the categories
##  <Ref Func="IsMultiplicativeElementWithInverse"/>
##  and <Ref Func="IsAssociativeElement"/>;
##  for a discussion of these implications,
##  see&nbsp;<Ref Sect="Arithmetic Operations for General Mappings"/>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategory( "IsGeneralMapping",
    IsMultiplicativeElementWithInverse and IsAssociativeElement );


#############################################################################
##
#C  IsSPGeneralMapping( <map> )
#C  IsNonSPGeneralMapping( <map> )
##
##  <#GAPDoc Label="IsSPGeneralMapping">
##  <ManSection>
##  <Filt Name="IsSPGeneralMapping" Arg='map' Type='Category'/>
##  <Filt Name="IsNonSPGeneralMapping" Arg='map' Type='Category'/>
##
##  <Description>
##  <!--  What we want to express is that <C>IsGeneralMapping</C> is the disjoint union-->
##  <!--  of <C>IsSPGeneralMapping</C> and <C>IsNonSPGeneralMapping</C>.-->
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategory( "IsSPGeneralMapping", IsGeneralMapping );
DeclareCategory( "IsNonSPGeneralMapping", IsGeneralMapping );


#############################################################################
##
#C  IsGeneralMappingCollection( <obj> )
##
##  <ManSection>
##  <Filt Name="IsGeneralMappingCollection" Arg='obj' Type='Category'/>
##
##  <Description>
##  </Description>
##  </ManSection>
##
DeclareCategoryCollections( "IsGeneralMapping" );


#############################################################################
##
#C  IsGeneralMappingFamily( <obj> )
##
##  <#GAPDoc Label="IsGeneralMappingFamily">
##  <ManSection>
##  <Filt Name="IsGeneralMappingFamily" Arg='obj' Type='Category'/>
##
##  <Description>
##  The family category of the category of general mappings.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategoryFamily( "IsGeneralMapping" );


#############################################################################
##
#A  FamilyRange( <Fam> )
##
##  <#GAPDoc Label="FamilyRange">
##  <ManSection>
##  <Attr Name="FamilyRange" Arg='Fam'/>
##
##  <Description>
##  is the elements family of the family of the range of each general
##  mapping in the family <A>Fam</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "FamilyRange", IsGeneralMappingFamily );


#############################################################################
##
#A  FamilySource( <Fam> )
##
##  <#GAPDoc Label="FamilySource">
##  <ManSection>
##  <Attr Name="FamilySource" Arg='Fam'/>
##
##  <Description>
##  is the elements family of the family of the source of each general
##  mapping in the family <A>Fam</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "FamilySource", IsGeneralMappingFamily );


#############################################################################
##
#A  FamiliesOfGeneralMappingsAndRanges( <Fam> )
##
##  <#GAPDoc Label="FamiliesOfGeneralMappingsAndRanges">
##  <ManSection>
##  <Attr Name="FamiliesOfGeneralMappingsAndRanges" Arg='Fam'/>
##
##  <Description>
##  is a list that stores at the odd positions the families of general
##  mappings with source in the family <A>Fam</A>, at the even positions the
##  families of ranges of the general mappings.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "FamiliesOfGeneralMappingsAndRanges",
    IsFamily, "mutable" );


#############################################################################
##
#P  IsConstantTimeAccessGeneralMapping( <map> )
##
##  <#GAPDoc Label="IsConstantTimeAccessGeneralMapping">
##  <ManSection>
##  <Prop Name="IsConstantTimeAccessGeneralMapping" Arg='map'/>
##
##  <Description>
##  is <K>true</K> if the underlying relation of the general mapping
##  <A>map</A> knows its <Ref Func="AsList"/> value,
##  and <K>false</K> otherwise.
##  <P/>
##  In the former case, <A>map</A> is allowed to use this list for calls to
##  <Ref Func="ImagesElm"/> etc.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsConstantTimeAccessGeneralMapping", IsGeneralMapping );


#############################################################################
##
#P  IsEndoGeneralMapping( <obj> )
##
##  <#GAPDoc Label="IsEndoGeneralMapping">
##  <ManSection>
##  <Prop Name="IsEndoGeneralMapping" Arg='obj'/>
##
##  <Description>
##  If a general mapping has this property then its source and range are
##  equal.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsEndoGeneralMapping", IsGeneralMapping );

#############################################################################
##
#P  IsTotal( <map> )  . . . . . . . . test whether a general mapping is total
##
##  <#GAPDoc Label="IsTotal">
##  <ManSection>
##  <Prop Name="IsTotal" Arg='map'/>
##
##  <Description>
##  is <K>true</K> if each element in the source <M>S</M>
##  of the general mapping <A>map</A> has images, i.e.,
##  <M>s^{<A>map</A>} \neq \emptyset</M> for all <M>s \in S</M>,
##  and <K>false</K> otherwise.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsTotal", IsGeneralMapping );


#############################################################################
##
#P  IsSingleValued( <map> ) . test whether a general mapping is single-valued
##
##  <#GAPDoc Label="IsSingleValued">
##  <ManSection>
##  <Prop Name="IsSingleValued" Arg='map'/>
##
##  <Description>
##  is <K>true</K> if each element in the source <M>S</M>
##  of the general mapping <A>map</A> has at most one image, i.e.,
##  <M>|s^{<A>map</A>}| \leq 1</M> for all <M>s \in S</M>,
##  and <K>false</K> otherwise.
##  <P/>
##  Equivalently, <C>IsSingleValued( <A>map</A> )</C> is <K>true</K>
##  if and only if the preimages of different elements in <M>R</M> are
##  disjoint.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsSingleValued", IsGeneralMapping );


#############################################################################
##
#P  IsMapping( <map> )
##
##  <#GAPDoc Label="IsMapping">
##  <ManSection>
##  <Filt Name="IsMapping" Arg='map'/>
##
##  <Description>
##  A <E>mapping</E> <A>map</A> is a general mapping that assigns to each
##  element <C>elm</C> of its source a unique element
##  <C>Image( <A>map</A>, elm )</C> of its range.
##  <P/>
##  Equivalently, the general mapping <A>map</A> is a mapping if and only if
##  it is total and single-valued
##  (see&nbsp;<Ref Func="IsTotal"/>, <Ref Func="IsSingleValued"/>).
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareSynonymAttr( "IsMapping",
    IsGeneralMapping and IsTotal and IsSingleValued );



#############################################################################
##
#P  IsEndoMapping( <obj> )
##
##  <ManSection>
##  <Prop Name="IsEndoMapping" Arg='obj'/>
##
##  <Description>
##  If a mapping has this property then its source and range are
##  equal and it is single valued.
##  </Description>
##  </ManSection>
##
DeclareSynonymAttr( "IsEndoMapping", IsMapping and IsEndoGeneralMapping );


#############################################################################
##
#P  IsInjective( <map> )  . . . . . .  test if a general mapping is injective
##
##  <#GAPDoc Label="IsInjective">
##  <ManSection>
##  <Prop Name="IsInjective" Arg='map'/>
##
##  <Description>
##  is <K>true</K> if the images of different elements in the source <M>S</M>
##  of the general mapping <A>map</A> are disjoint, i.e.,
##  <M>x^{<A>map</A>} \cap y^{<A>map</A>} = \emptyset</M>
##  for <M>x \neq y \in S</M>,
##  and <K>false</K> otherwise.
##  <P/>
##  Equivalently, <C>IsInjective( <A>map</A> )</C> is <K>true</K>
##  if and only if each element in the range of <A>map</A> has at most one
##  preimage in <M>S</M>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsInjective", IsGeneralMapping );
DeclareSynonym("IsOneToOne",IsInjective);

#############################################################################
##
#P  IsSurjective( <map> ) . . . . . . test if a general mapping is surjective
##
##  <#GAPDoc Label="IsSurjective">
##  <ManSection>
##  <Prop Name="IsSurjective" Arg='map'/>
##
##  <Description>
##  is <K>true</K> if each element in the range <M>R</M>
##  of the general mapping <A>map</A> has preimages in the source <M>S</M>
##  of <A>map</A>, i.e.,
##  <M>\{ s \in S \mid x \in s^{<A>map</A>} \} \neq \emptyset</M>
##  for all <M>x \in R</M>, and <K>false</K> otherwise.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsSurjective", IsGeneralMapping );
DeclareSynonym("IsOnto",IsSurjective);


#############################################################################
##
#P  IsBijective( <map> )  . . . . . .  test if a general mapping is bijective
##
##  <#GAPDoc Label="IsBijective">
##  <ManSection>
##  <Prop Name="IsBijective" Arg='map'/>
##
##  <Description>
##  A general mapping <A>map</A> is <E>bijective</E> if and only if it is
##  an injective and surjective mapping (see&nbsp;<Ref Func="IsMapping"/>,
##  <Ref Func="IsInjective"/>, <Ref Func="IsSurjective"/>).
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareSynonymAttr( "IsBijective",
    IsSingleValued and IsTotal and IsInjective and IsSurjective );


#############################################################################
##
#A  Range( <map> )  . . . . . . . . . . . . . . .  range of a general mapping
##
##  <#GAPDoc Label="Range">
##  <ManSection>
##  <Attr Name="Range" Arg='map' Label="of a general mapping"/>
##
##  <Description>
##  The range of a general mapping.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "Range", IsGeneralMapping );


#############################################################################
##
#A  Source( <map> ) . . . . . . . . . . . . . . . source of a general mapping
##
##  <#GAPDoc Label="Source">
##  <ManSection>
##  <Attr Name="Source" Arg='map'/>
##
##  <Description>
##  The source of a general mapping.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "Source", IsGeneralMapping );


#############################################################################
##
#A  UnderlyingRelation( <map> ) . .  underlying relation of a general mapping
##
##  <#GAPDoc Label="UnderlyingRelation">
##  <ManSection>
##  <Attr Name="UnderlyingRelation" Arg='map'/>
##
##  <Description>
##  The <E>underlying relation</E> of a general mapping <A>map</A> is the
##  domain of pairs <M>(s,r)</M>, with <M>s</M> in the source and <M>r</M> in
##  the range of <A>map</A> (see&nbsp;<Ref Func="Source"/>,
##  <Ref Func="Range" Label="of a general mapping"/>),
##  and <M>r \in</M> <C>ImagesElm( <A>map</A>, </C><M>s</M><C> )</C>.
##  <P/>
##  Each element of the underlying relation is represented by
##  a direct product element (see&nbsp;<Ref Func="IsDirectProductElement"/>).
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "UnderlyingRelation", IsGeneralMapping );


#############################################################################
##
#A  UnderlyingGeneralMapping( <map> )
##
##  <#GAPDoc Label="UnderlyingGeneralMapping">
##  <ManSection>
##  <Attr Name="UnderlyingGeneralMapping" Arg='map'/>
##
##  <Description>
##  attribute for underlying relations of general mappings
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "UnderlyingGeneralMapping", IsCollection );


#############################################################################
##
#F  GeneralMappingsFamily( <sourcefam>, <rangefam> )
##
##  <#GAPDoc Label="GeneralMappingsFamily">
##  <ManSection>
##  <Func Name="GeneralMappingsFamily" Arg='sourcefam, rangefam'/>
##
##  <Description>
##  All general mappings with same source family <A>FS</A> and same range
##  family <A>FR</A> lie in the family
##  <C>GeneralMappingsFamily( <A>FS</A>, <A>FR</A> )</C>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "GeneralMappingsFamily" );


#############################################################################
##
#F  TypeOfDefaultGeneralMapping( <source>, <range>, <filter> )
##
##  <#GAPDoc Label="TypeOfDefaultGeneralMapping">
##  <ManSection>
##  <Func Name="TypeOfDefaultGeneralMapping" Arg='source, range, filter'/>
##
##  <Description>
##  is the type of mappings with <C>IsDefaultGeneralMappingRep</C> with
##  source <A>source</A> and range <A>range</A> and additional categories
##  <A>filter</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "TypeOfDefaultGeneralMapping" );


#############################################################################
##
#A  IdentityMapping( <D> )  . . . . . . . .  identity mapping of a collection
##
##  <#GAPDoc Label="IdentityMapping">
##  <ManSection>
##  <Attr Name="IdentityMapping" Arg='D'/>
##
##  <Description>
##  is the bijective mapping with source and range equal to the collection
##  <A>D</A>, which maps each element of <A>D</A> to itself.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "IdentityMapping", IsCollection );


#############################################################################
##
#A  InverseGeneralMapping( <map> )
##
##  <#GAPDoc Label="InverseGeneralMapping">
##  <ManSection>
##  <Attr Name="InverseGeneralMapping" Arg='map'/>
##
##  <Description>
##  The <E>inverse general mapping</E> of a general mapping <A>map</A> is
##  the general mapping whose underlying relation
##  (see&nbsp;<Ref Func="UnderlyingRelation"/>) contains a pair <M>(r,s)</M>
##  if and only if the underlying relation of <A>map</A> contains the pair
##  <M>(s,r)</M>.
##  <P/>
##  See the introduction to Chapter&nbsp;<Ref Chap="Mappings"/>
##  for the subtleties concerning the difference between
##  <Ref Func="InverseGeneralMapping"/> and <Ref Func="Inverse"/>.
##  <P/>
##  Note that the inverse general mapping of a mapping <A>map</A> is
##  in general only a general mapping.
##  If <A>map</A> knows to be bijective its inverse general mapping will know
##  to be a mapping.
##  In this case also <C>Inverse( <A>map</A> )</C> works.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "InverseGeneralMapping", IsGeneralMapping );


#############################################################################
##
#A  ImagesSource( <map> )
##
##  <#GAPDoc Label="ImagesSource">
##  <ManSection>
##  <Attr Name="ImagesSource" Arg='map'/>
##
##  <Description>
##  is the set of images of the source of the general mapping <A>map</A>.
##  <P/>
##  <Ref Func="ImagesSource"/> delegates to <Ref Func="ImagesSet"/>,
##  it is introduced only to store the image of <A>map</A> as attribute
##  value.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "ImagesSource", IsGeneralMapping );


#############################################################################
##
#A  PreImagesRange( <map> )
##
##  <#GAPDoc Label="PreImagesRange">
##  <ManSection>
##  <Attr Name="PreImagesRange" Arg='map'/>
##
##  <Description>
##  is the set of preimages of the range of the general mapping <A>map</A>.
##  <P/>
##  <Ref Func="PreImagesRange"/> delegates to <Ref Func="PreImagesSet"/>,
##  it is introduced only to store the preimage of <A>map</A> as attribute
##  value.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "PreImagesRange", IsGeneralMapping );


#############################################################################
##
#O  ImagesElm( <map>, <elm> ) . . . all images of an elm under a gen. mapping
##
##  <#GAPDoc Label="ImagesElm">
##  <ManSection>
##  <Oper Name="ImagesElm" Arg='map, elm'/>
##
##  <Description>
##  If <A>elm</A> is an element of the source of the general mapping
##  <A>map</A> then <Ref Func="ImagesElm"/> returns the set of all images
##  of <A>elm</A> under <A>map</A>.
##  <P/>
##  Anything may happen if <A>elm</A> is not an element of the source of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "ImagesElm", [ IsGeneralMapping, IsObject ] );


#############################################################################
##
#O  ImagesRepresentative(<map>,<elm>) . one image of elm under a gen. mapping
##
##  <#GAPDoc Label="ImagesRepresentative">
##  <ManSection>
##  <Oper Name="ImagesRepresentative" Arg='map,elm'/>
##
##  <Description>
##  If <A>elm</A> is an element of the source of the general mapping
##  <A>map</A> then <Ref Func="ImagesRepresentative"/> returns either
##  a representative of the set of images of <A>elm</A> under <A>map</A>
##  or <K>fail</K>, the latter if and only if <A>elm</A> has no images under
##  <A>map</A>.
##  <P/>
##  Anything may happen if <A>elm</A> is not an element of the source of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "ImagesRepresentative", [ IsGeneralMapping, IsObject ] );


#############################################################################
##
#O  ImagesSet( <map>, <elms> )
##
##  <#GAPDoc Label="ImagesSet">
##  <ManSection>
##  <Oper Name="ImagesSet" Arg='map, elms'/>
##
##  <Description>
##  If <A>elms</A> is a subset of the source of the general mapping
##  <A>map</A> then <Ref Func="ImagesSet"/> returns the set of all images of
##  <A>elms</A> under <A>map</A>.
##  <P/>
##  The result will be either a proper set or a domain.
##  Anything may happen if <A>elms</A> is not a subset of the source of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "ImagesSet", [ IsGeneralMapping, IsCollection ] );


#############################################################################
##
#O  ImageElm( <map>, <elm> )  . . . .  unique image of an elm under a mapping
##
##  <#GAPDoc Label="ImageElm">
##  <ManSection>
##  <Oper Name="ImageElm" Arg='map, elm'/>
##
##  <Description>
##  If <A>elm</A> is an element of the source of the total and single-valued
##  mapping <A>map</A> then
##  <Ref Func="ImageElm"/> returns the unique image of <A>elm</A> under
##  <A>map</A>.
##  <P/>
##  Anything may happen if <A>elm</A> is not an element of the source of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "ImageElm", [ IsMapping, IsObject ] );


#############################################################################
##
#F  Image( <map> )  . . . .  set of images of the source of a general mapping
#F  Image( <map>, <elm> ) . . . .  unique image of an element under a mapping
#F  Image( <map>, <coll> )  . . set of images of a collection under a mapping
##
##  <#GAPDoc Label="Image">
##  <ManSection>
##  <Heading>Image</Heading>
##  <Func Name="Image" Arg='map'
##   Label="set of images of the source of a general mapping"/>
##  <Func Name="Image" Arg='map, elm'
##   Label="unique image of an element under a mapping"/>
##  <Func Name="Image" Arg='map, coll'
##   Label="set of images of a collection under a mapping"/>
##
##  <Description>
##  <C>Image( <A>map</A> )</C> is the <E>image</E> of the general mapping
##  <A>map</A>, i.e.,
##  the subset of elements of the range of <A>map</A>
##  that are actually values of <A>map</A>.
##  <E>Note</E> that in this case the argument may also be multi-valued.
##  <P/>
##  <C>Image( <A>map</A>, <A>elm</A> )</C> is the image of the element
##  <A>elm</A> of the source of the mapping <A>map</A> under <A>map</A>,
##  i.e., the unique element of the range to which <A>map</A> maps
##  <A>elm</A>.
##  This can also be expressed as <A>elm</A><C>^</C><A>map</A>.
##  Note that <A>map</A> must be total and single valued,
##  a multi valued general mapping is not allowed
##  (see&nbsp;<Ref Func="Images" Label="set of images of the source of a general mapping"/>).
##  <P/>
##  <C>Image( <A>map</A>, <A>coll</A> )</C> is the image of the subset
##  <A>coll</A> of the source of the mapping <A>map</A> under <A>map</A>,
##  i.e., the subset of the range to which <A>map</A> maps elements of
##  <A>coll</A>.
##  <A>coll</A> may be a proper set or a domain.
##  The result will be either a proper set or a domain.
##  Note that in this case <A>map</A> may also be multi-valued.
##  (If <A>coll</A> and the result are lists then the positions of
##  entries do in general <E>not</E> correspond.)
##  <P/>
##  <Ref Func="Image" Label="set of images of the source of a general mapping"/>
##  delegates to <Ref Func="ImagesSource"/> when called
##  with one argument, and to <Ref Func="ImageElm"/> resp.
##  <Ref Func="ImagesSet"/> when called with two arguments.
##  <P/>
##  If the second argument is not an element or a subset of the source of
##  the first argument, an error is signalled.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Image" );


#############################################################################
##
#F  Images( <map> ) . . . .  set of images of the source of a general mapping
#F  Images( <map>, <elm> )  . . . set of images of an element under a mapping
#F  Images( <map>, <coll> ) . . set of images of a collection under a mapping
##
##  <#GAPDoc Label="Images">
##  <ManSection>
##  <Heading>Images</Heading>
##  <Func Name="Images" Arg='map'
##   Label="set of images of the source of a general mapping"/>
##  <Func Name="Images" Arg='map, elm'
##   Label="set of images of an element under a mapping"/>
##  <Func Name="Images" Arg='map, coll'
##   Label="set of images of a collection under a mapping"/>
##
##  <Description>
##  <C>Images( <A>map</A> )</C> is the <E>image</E> of the general mapping
##  <A>map</A>, i.e., the subset of elements of the range of <A>map</A>
##  that are actually values of <A>map</A>.
##  <P/>
##  <C>Images( <A>map</A>, <A>elm</A> )</C> is the set of images of the
##  element <A>elm</A> of the source of the general mapping <A>map</A> under
##  <A>map</A>, i.e., the set of elements of the range to which <A>map</A>
##  maps <A>elm</A>.
##  <P/>
##  <C>Images( <A>map</A>, <A>coll</A> )</C> is the set of images of the
##  subset <A>coll</A> of the source of the general mapping <A>map</A> under
##  <A>map</A>, i.e., the subset of the range to which <A>map</A> maps
##  elements of <A>coll</A>.
##  <A>coll</A> may be a proper set or a domain.
##  The result will be either a proper set or a domain.
##  (If <A>coll</A> and the result are lists then the positions of
##  entries do in general <E>not</E> correspond.)
##  <P/>
##  <Ref Func="Images" Label="set of images of the source of a general mapping"/>
##  delegates to <Ref Func="ImagesSource"/> when called
##  with one argument, and to <Ref Func="ImagesElm"/> resp.
##  <Ref Func="ImagesSet"/> when called with two arguments.
##  <P/>
##  If the second argument is not an element or a subset of the source of
##  the first argument, an error is signalled.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Images" );


#############################################################################
##
#O  PreImagesElm( <map>, <elm> )  . all preimages of elm under a gen. mapping
##
##  <#GAPDoc Label="PreImagesElm">
##  <ManSection>
##  <Oper Name="PreImagesElm" Arg='map, elm'/>
##
##  <Description>
##  If <A>elm</A> is an element of the range of the general mapping
##  <A>map</A> then <Ref Func="PreImagesElm"/> returns the set of all
##  preimages of <A>elm</A> under <A>map</A>.
##  <P/>
##  Anything may happen if <A>elm</A> is not an element of the range of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "PreImagesElm", [ IsGeneralMapping, IsObject ] );


#############################################################################
##
#O  PreImageElm( <map>, <elm> )
##
##  <#GAPDoc Label="PreImageElm">
##  <ManSection>
##  <Oper Name="PreImageElm" Arg='map, elm'/>
##
##  <Description>
##  If <A>elm</A> is an element of the range of the injective and surjective
##  general mapping <A>map</A> then
##  <Ref Func="PreImageElm"/> returns the unique preimage of <A>elm</A> under
##  <A>map</A>.
##  <P/>
##  Anything may happen if <A>elm</A> is not an element of the range of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "PreImageElm",
    [ IsGeneralMapping and IsInjective and IsSurjective, IsObject ] );


#############################################################################
##
#O  PreImagesRepresentative( <map>, <elm> ) . . .  one preimage of an element
##                                                       under a gen. mapping
##
##  <#GAPDoc Label="PreImagesRepresentative">
##  <ManSection>
##  <Oper Name="PreImagesRepresentative" Arg='map, elm'/>
##
##  <Description>
##  If <A>elm</A> is an element of the range of the general mapping
##  <A>map</A> then <Ref Func="PreImagesRepresentative"/> returns either a
##  representative of the set of preimages of <A>elm</A> under <A>map</A> or
##  <K>fail</K>, the latter if and only if <A>elm</A>
##  has no preimages under <A>map</A>.
##  <P/>
##  Anything may happen if <A>elm</A> is not an element of the range of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "PreImagesRepresentative",
    [ IsGeneralMapping, IsObject ] );


#############################################################################
##
#O  PreImagesSet( <map>, <elms> )
##
##  <#GAPDoc Label="PreImagesSet">
##  <ManSection>
##  <Oper Name="PreImagesSet" Arg='map, elms'/>
##
##  <Description>
##  If <A>elms</A> is a subset of the range of the general mapping <A>map</A>
##  then <Ref Func="PreImagesSet"/> returns the set of all preimages of
##  <A>elms</A> under <A>map</A>.
##  <P/>
##  Anything may happen if <A>elms</A> is not a subset of the range of
##  <A>map</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "PreImagesSet", [ IsGeneralMapping, IsCollection ] );


#############################################################################
##
#F  PreImage( <map> ) . .  set of preimages of the range of a general mapping
#F  PreImage( <map>, <elm> )  . unique preimage of an elm under a gen.mapping
#F  PreImage(<map>, <coll>)  set of preimages of a coll. under a gen. mapping
##
##  <#GAPDoc Label="PreImage">
##  <ManSection>
##  <Heading>PreImage</Heading>
##  <Func Name="PreImage" Arg='map'
##   Label="set of preimages of the range of a general mapping"/>
##  <Func Name="PreImage" Arg='map, elm'
##   Label="unique preimage of an element under a general mapping"/>
##  <Func Name="PreImage" Arg='map, coll'
##   Label="set of preimages of a collection under a general mapping"/>
##
##  <Description>
##  <C>PreImage( <A>map</A> )</C> is the preimage of the general mapping
##  <A>map</A>, i.e., the subset of elements of the source of <A>map</A>
##  that actually have values under <A>map</A>.
##  Note that in this case the argument may also be non-injective or
##  non-surjective.
##  <P/>
##  <C>PreImage( <A>map</A>, <A>elm</A> )</C> is the preimage of the element
##  <A>elm</A> of the range of the injective and surjective mapping
##  <A>map</A> under <A>map</A>, i.e., the unique element of the source
##  which is mapped under <A>map</A> to <A>elm</A>.
##  Note that <A>map</A> must be injective and surjective
##  (see&nbsp;<Ref Func="PreImages" Label="set of preimages of the range of a general mapping"/>).
##  <P/>
##  <C>PreImage( <A>map</A>, <A>coll</A> )</C> is the preimage of the subset
##  <A>coll</A> of the range of the general mapping <A>map</A> under
##  <A>map</A>, i.e., the subset of the source which is mapped under
##  <A>map</A> to elements of <A>coll</A>. <A>coll</A> may be a proper set
##  or a domain.
##  The result will be either a proper set or a domain.
##  Note that in this case <A>map</A> may also be non-injective or
##  non-surjective.
##  (If <A>coll</A> and the result are lists then the positions of
##  entries do in general <E>not</E> correspond.)
##  <P/>
##  <Ref Func="PreImage" Label="set of preimages of the range of a general mapping"/>
##  delegates to <Ref Func="PreImagesRange"/> when
##  called with one argument,
##  and to <Ref Func="PreImageElm"/> resp. <Ref Func="PreImagesSet"/> when
##  called with two arguments.
##  <P/>
##  If the second argument is not an element or a subset of the range of
##  the first argument, an error is signalled.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "PreImage" );


#############################################################################
##
#F  PreImages( <map> )  . . . set of preimages of the range of a gen. mapping
#F  PreImages(<map>,<elm>)  . set of preimages of an elm under a gen. mapping
#F  PreImages(<map>,<coll>)  set of preimages of a coll. under a gen. mapping
##
##  <#GAPDoc Label="PreImages">
##  <ManSection>
##  <Heading>PreImages</Heading>
##  <Func Name="PreImages" Arg='map'
##   Label="set of preimages of the range of a general mapping"/>
##  <Func Name="PreImages" Arg='map, elm'
##   Label="set of preimages of an elm under a general mapping"/>
##  <Func Name="PreImages" Arg='map, coll'
##   Label="set of preimages of a collection under a general mapping"/>
##
##  <Description>
##  <C>PreImages( <A>map</A> )</C> is the preimage of the general mapping
##  <A>map</A>, i.e., the subset of elements of the source of <A>map</A>
##  that have actually values under <A>map</A>.
##  <P/>
##  <C>PreImages( <A>map</A>, <A>elm</A> )</C> is the set of preimages of the
##  element <A>elm</A> of the range of the general mapping <A>map</A> under
##  <A>map</A>, i.e., the set of elements of the source which <A>map</A> maps
##  to <A>elm</A>.
##  <P/>
##  <C>PreImages( <A>map</A>, <A>coll</A> )</C> is the set of images of the
##  subset <A>coll</A> of the range of the general mapping <A>map</A> under
##  <A>map</A>, i.e., the subset of the source which <A>map</A> maps to
##  elements of <A>coll</A>.
##  <A>coll</A> may be a proper set or a domain.
##  The result will be either a proper set or a domain.
##  (If <A>coll</A> and the result are lists then the positions of
##  entries do in general <E>not</E> correspond.)
##  <P/>
##  <Ref Func="PreImages" Label="set of preimages of the range of a general mapping"/>
##  delegates to <Ref Func="PreImagesRange"/> when
##  called with one argument,
##  and to <Ref Func="PreImagesElm"/> resp. <Ref Func="PreImagesSet"/> when
##  called with two arguments.
##  <P/>
##  If the second argument is not an element or a subset of the range of
##  the first argument, an error is signalled.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "PreImages" );


#############################################################################
##
#O  CompositionMapping2(<map2>,<map1>)  . . . composition of general mappings
#F  CompositionMapping2General(<map2>,<map1>)
##
##  <#GAPDoc Label="CompositionMapping2">
##  <ManSection>
##  <Oper Name="CompositionMapping2" Arg='map2, map1'/>
##  <Func Name="CompositionMapping2General" Arg='map2, map1'/>
##
##  <Description>
##  <Ref Func="CompositionMapping2"/> returns the composition of <A>map2</A>
##  and <A>map1</A>,
##  this is the general mapping that maps an element first under <A>map1</A>,
##  and then maps the images under <A>map2</A>.
##  <P/>
##  (Note the reverse ordering of arguments in the composition via
##  the multiplication <Ref Func="\*"/>.
##  <P/>
##  <Ref Func="CompositionMapping2General"/> is the method that forms a
##  composite mapping with two constituent mappings.
##  (This is used in some algorithms.)
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "CompositionMapping2",
    [ IsGeneralMapping, IsGeneralMapping ] );

DeclareGlobalFunction("CompositionMapping2General");


#############################################################################
##
#F  CompositionMapping( <map1>, <map2>, ... ) . . . . composition of mappings
##
##  <#GAPDoc Label="CompositionMapping">
##  <ManSection>
##  <Func Name="CompositionMapping" Arg='map1, map2, ...'/>
##
##  <Description>
##  <Ref Func="CompositionMapping"/> allows one to compose arbitrarily many
##  general mappings,
##  and delegates each step to <Ref Func="CompositionMapping2"/>.
##  <P/>
##  Additionally, the properties <Ref Func="IsInjective"/> and
##  <Ref Func="IsSingleValued"/> are maintained;
##  if the source of the <M>i+1</M>-th general mapping is identical to
##  the range of the <M>i</M>-th general mapping,
##  also <Ref Func="IsTotal"/> and <Ref Func="IsSurjective"/> are maintained.
##  (So one should not call <Ref Func="CompositionMapping2"/> directly
##  if one wants to maintain these properties.)
##  <P/>
##  Depending on the types of <A>map1</A> and <A>map2</A>,
##  the returned mapping might be constructed completely new (for example by
##  giving domain generators and their images, this is for example the case
##  if both mappings preserve the same algebraic structures and &GAP; can
##  decompose elements of the source of <A>map2</A> into generators) or as an
##  (iterated) composition
##  (see&nbsp;<Ref Func="IsCompositionMappingRep"/>).
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "CompositionMapping" );


#############################################################################
##
#R  IsCompositionMappingRep( <map> )
##
##  <#GAPDoc Label="IsCompositionMappingRep">
##  <ManSection>
##  <Filt Name="IsCompositionMappingRep" Arg='map' Type='Representation'/>
##
##  <Description>
##  Mappings in this representation are stored as composition of two
##  mappings, (pre)images of elements are computed in a two-step process.
##  The constituent mappings of the composition can be obtained via
##  <Ref Func="ConstituentsCompositionMapping"/>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareRepresentation( "IsCompositionMappingRep",
    IsGeneralMapping and IsAttributeStoringRep, [ "map1", "map2" ] );


#############################################################################
##
#F  ConstituentsCompositionMapping( <map> )
##
##  <#GAPDoc Label="ConstituentsCompositionMapping">
##  <ManSection>
##  <Func Name="ConstituentsCompositionMapping" Arg='map'/>
##
##  <Description>
##  If <A>map</A> is stored in the representation
##  <Ref Func="IsCompositionMappingRep"/> as composition of two mappings
##  <A>map1</A> and <A>map2</A>, this function returns the
##  two constituent mappings in a list <C>[ <A>map1</A>, <A>map2</A> ]</C>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "ConstituentsCompositionMapping" );


#############################################################################
##
#O  ZeroMapping( <S>, <R> ) . . . . . . . . . .  zero mapping from <S> to <R>
##
##  <#GAPDoc Label="ZeroMapping">
##  <ManSection>
##  <Oper Name="ZeroMapping" Arg='S, R'/>
##
##  <Description>
##  A zero mapping is a total general mapping that maps each element of its
##  source to the zero element of its range.
##  <P/>
##  (Each mapping with empty source is a zero mapping.)
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "ZeroMapping", [ IsCollection, IsCollection ] );


#############################################################################
##
#O  RestrictedMapping( <map>, <subdom> )
##
##  <#GAPDoc Label="RestrictedMapping">
##  <ManSection>
##  <Oper Name="RestrictedMapping" Arg='map, subdom'/>
##
##  <Description>
##  If <A>subdom</A> is a subdomain of the source of the general mapping
##  <A>map</A>,
##  this operation returns the restriction of <A>map</A> to <A>subdom</A>.
##  <!--  The general concept of restricted general mappings still missing.-->
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "RestrictedMapping", [ IsGeneralMapping, IsDomain ] );


#############################################################################
##
#R  IsGeneralRestrictedMappingRep( <map> )
##
##  Mappings in this representation are stored as wrapper object, containing
##  the original map but new source and range.
##
DeclareRepresentation( "IsGeneralRestrictedMappingRep",
    IsGeneralMapping and IsAttributeStoringRep, [ "map" ] );

#############################################################################
##
#F  GeneralRestrictedMapping( <map>, <source>, <range> )
##
##  <C>GeneralRestrictedMapping</C> allows one to restrict <source> and
##  <range> for an existing mapping, for example enforcing injectivity or
##  surjectivity this way.
##
DeclareGlobalFunction( "GeneralRestrictedMapping" );

#############################################################################
##
#O  Embedding( <S>, <T> ) . . . . . . .  embedding of one domain into another
#O  Embedding( <S>, <i> )
##
##  <#GAPDoc Label="Embedding">
##  <ManSection>
##  <Heading>Embedding</Heading>
##  <Oper Name="Embedding" Arg='S, T' Label="for two domains"/>
##  <Oper Name="Embedding" Arg='S, i'
##   Label="for a domain and a positive integer"/>
##
##  <Description>
##  returns the embedding of the domain <A>S</A> in the domain <A>T</A>,
##  or in the second form, some domain indexed by the positive integer
##  <A>i</A>.
##  The precise natures of the various methods are described elsewhere:
##  for Lie algebras, see <Ref Func="LieFamily"/>; for group  products,
##  see&nbsp;<Ref Sect="Embeddings and Projections for Group Products"/>
##  for a general description, or for examples
##  see&nbsp;<Ref Sect="Direct Products"/> for direct products,
##  <Ref Sect="Semidirect Products"/> for semidirect products,
##  or&nbsp;<Ref Sect="Wreath Products"/> for wreath products; or for
##  magma rings
##  see&nbsp;<Ref Sect="Natural Embeddings related to Magma Rings"/>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "Embedding", [ IsDomain, IsObject ] );


#############################################################################
##
#O  Projection( <S>, <T> )  . . . . . . projection of one domain onto another
#O  Projection( <S>, <i> )
#O  Projection( <S> )
##
##  <#GAPDoc Label="Projection">
##  <ManSection>
##  <Heading>Projection</Heading>
##  <Oper Name="Projection" Arg='S, T' Label="for two domains"/>
##  <Oper Name="Projection" Arg='S, i'
##   Label="for a domain and a positive integer"/>
##  <Oper Name="Projection" Arg='S' Label="for a domain"/>
##
##  <Description>
##  returns the projection of the domain <A>S</A> onto the domain <A>T</A>,
##  or in the second form, some domain indexed by the positive integer
##  <A>i</A>,
##  or in the third form some natural quotient domain of <A>S</A>.
##  Various methods are defined for group products;
##  see&nbsp;<Ref Sect="Embeddings and Projections for Group Products"/> for
##  a general description,
##  or for examples see&nbsp;<Ref Sect="Direct Products"/> for direct
##  products, <Ref Sect="Semidirect Products"/> for semidirect products,
##  <Ref Sect="Subdirect Products"/> for subdirect products,
##  or&nbsp;<Ref Sect="Wreath Products"/> for wreath products.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "Projection", [ IsDomain, IsObject ] );


#############################################################################
##
#F  GeneralMappingByElements( <S>, <R>, <elms> )
##
##  <#GAPDoc Label="GeneralMappingByElements">
##  <ManSection>
##  <Func Name="GeneralMappingByElements" Arg='S, R, elms'/>
##
##  <Description>
##  is the general mapping with source <A>S</A> and range <A>R</A>,
##  and with underlying relation consisting of the collection <A>elms</A>
##  of direct product elements.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "GeneralMappingByElements" );


#############################################################################
##                                         
#F  MappingByFunction( <S>, <R>, <fun>[, <invfun>] )
#F  MappingByFunction( <S>, <R>, <fun>, `false', <prefun> )
##
##  <#GAPDoc Label="MappingByFunction">
##  <ManSection>
##  <Heading>MappingByFunction</Heading>
##  <Func Name="MappingByFunction" Arg='S, R, fun[, invfun]'
##   Label="by function (and inverse function) between two domains"/>
##  <Func Name="MappingByFunction" Arg='S, R, fun, false, prefun'
##   Label="by function and function that computes one preimage"/>
##
##  <Description>
##  <Ref Func="MappingByFunction" Label="by function (and inverse function) between two domains"/>
##  returns a mapping <C>map</C> with source
##  <A>S</A> and range <A>R</A>,
##  such that each element <M>s</M> of <A>S</A> is mapped to the element
##  <A>fun</A><M>( s )</M>, where <A>fun</A> is a &GAP; function.
##  <P/>
##  If the argument <A>invfun</A> is bound then <C>map</C> is a bijection
##  between <A>S</A> and <A>R</A>, and the preimage of each element <M>r</M>
##  of <A>R</A> is given by <A>invfun</A><M>( r )</M>,
##  where <A>invfun</A> is a &GAP;  function.
##  <P/>
##  If five arguments are given and the fourth argument is <K>false</K> then
##  the &GAP; function <A>prefun</A> can be used to compute a single preimage
##  also if <C>map</C> is not bijective.
##  <!-- what is <A>prefun</A> expected to return for <A>r</A> outside the image of <A>map</A>-->
##  <!-- if <A>map</A> is not surjective?-->
##  <!-- or must <A>map</A> be surjective in this case?-->
##  <P/>
##  The mapping returned by
##  <Ref Func="MappingByFunction" Label="by function (and inverse function) between two domains"/> lies in the
##  filter <Ref Func="IsNonSPGeneralMapping"/>,
##  see&nbsp;<Ref Sect="Technical Matters Concerning General Mappings"/>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "MappingByFunction" );


#############################################################################
##
#m  IsBijective . . . . . . . . . . . . . . . . . . . .  for identity mapping
##
InstallTrueMethod( IsBijective, IsGeneralMapping and IsOne );


#############################################################################
##
#m  IsSingleValued  . . . . . . . . . . . . . . . . . . . .  for zero mapping
#m  IsTotal . . . . . . . . . . . . . . . . . . . . . . . .  for zero mapping
##
InstallTrueMethod( IsSingleValued, IsGeneralMapping and IsZero );
InstallTrueMethod( IsTotal, IsGeneralMapping and IsZero );


#############################################################################
##
#F  CopyMappingAttributes( <from>, <to> )
##
##  <ManSection>
##  <Func Name="CopyMappingAttributes" Arg='from, to'/>
##
##  <Description>
##  Let <A>from</A> and <A>to</A> be two general mappings which are known to be equal.
##  <C>CopyMappingAttributes</C> copies known mapping attributes from <A>from</A> to
##  <A>to</A>. This is used in operations, such as
##  <C>AsGroupGeneralMappingByImages</C>, that produce equal mappings in another
##  representation.
##  </Description>
##  </ManSection>
##
DeclareGlobalFunction( "CopyMappingAttributes" );

#############################################################################
##
#A  MappingGeneratorsImages(<map>)
##
##  <#GAPDoc Label="MappingGeneratorsImages">
##  <ManSection>
##  <Attr Name="MappingGeneratorsImages" Arg='map'/>
##
##  <Description>
##  This attribute contains a list of length 2, the first entry being a list
##  of generators of the source of <A>map</A> and the second entry a list of
##  their images. This attribute is used, for example, by
##  <Ref Func="GroupHomomorphismByImages"/> to store generators and images.
##  <!--  <C>MappingGeneratorsImages</C> is permitted to call           -->
##  <!--  <C>Source</C> and <C>ImagesRepresentative</C>.                -->
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "MappingGeneratorsImages", IsGeneralMapping );


#############################################################################
##
#E