This file is indexed.

/usr/share/gap/doc/ref/chap32.html is in gap-doc 4r8p8-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  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
<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 32: Mappings</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap32"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap31.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap33.html">[Next Chapter]</a>&nbsp;  </div>

<p id="mathjaxlink" class="pcenter"><a href="chap32_mj.html">[MathJax on]</a></p>
<p><a id="X7C9734B880042C73" name="X7C9734B880042C73"></a></p>
<div class="ChapSects"><a href="chap32.html#X7C9734B880042C73">32 <span class="Heading">Mappings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X783BAB2683EEA0CC">32.1 <span class="Heading">IsDirectProductElement (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X87FD9FE787023FF0">32.1-1 IsDirectProductElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X7CF6FEFB8290D5CB">32.2 <span class="Heading">Creating Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X79D0D2F07A14D039">32.2-1 GeneralMappingByElements</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7D55E1977ED70E01">32.2-2 <span class="Heading">MappingByFunction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X865FC25A87D36F3D">32.2-3 InverseGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7ED1E4E27CCE2DCA">32.2-4 CompositionMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X86486B687B7077AC">32.2-5 CompositionMapping2</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7A926D167C3155F6">32.2-6 IsCompositionMappingRep</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X87775B438008DCA5">32.2-7 ConstituentsCompositionMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X795FF8DC785F110A">32.2-8 ZeroMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7EBAE0368470A603">32.2-9 IdentityMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X86452F8587CBAEA0">32.2-10 <span class="Heading">Embedding</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X8769E8DA80BC96C1">32.2-11 <span class="Heading">Projection</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X800014D683A81009">32.2-12 RestrictedMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X7E5A430D7F838F1C">32.3 <span class="Heading">Properties and Attributes of (General) Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X83C7494E828CC9C8">32.3-1 IsTotal</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X86D44C8A78BF1981">32.3-2 IsSingleValued</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7CC95EB282854385">32.3-3 IsMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7F065FD7822C0A12">32.3-4 IsInjective</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X784ECE847E005B8F">32.3-5 IsSurjective</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X878F56AB7B342767">32.3-6 IsBijective</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7B6FD7277CDE9FCB">32.3-7 Range</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7DE8173F80E07AB1">32.3-8 Source</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X784F871383FB599B">32.3-9 UnderlyingRelation</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X786581DE871A47D0">32.3-10 UnderlyingGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X83B4FF15847F06FC">32.4 <span class="Heading">Images under Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7D23C1CE863DACD8">32.4-1 ImagesSource</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X85ADB89B7C8DD7D0">32.4-2 ImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7D51184B7EE5B2CF">32.4-3 ImagesElm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X8781348F7F5796A0">32.4-4 ImagesSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7CFAB0157BFB1806">32.4-5 ImageElm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X87F4D35A826599C6">32.4-6 <span class="Heading">Image</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X86114B2E7E77488C">32.4-7 <span class="Heading">Images</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X79BB1EC07C828667">32.5 <span class="Heading">Preimages under Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X78EF1FE77B0973C0">32.5-1 PreImagesRange</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7FBB830C8729E995">32.5-2 PreImagesElm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7D212F727CAE971A">32.5-3 PreImageElm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7AE24A1586B7DE79">32.5-4 PreImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X856BAFC87B2D2811">32.5-5 PreImagesSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X836FAEAC78B55BF4">32.5-6 <span class="Heading">PreImage</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X85C8590E832002EF">32.5-7 <span class="Heading">PreImages</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X7E2E16277940FA0B">32.6 <span class="Heading">Arithmetic Operations for General Mappings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X834E02BB7D4B4AE5">32.7 <span class="Heading">Mappings which are Compatible with Algebraic Structures</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X8008FCCC7F4C731F">32.8 <span class="Heading">Magma Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7DC72CF28539A251">32.8-1 IsMagmaHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X8181676787E760A2">32.8-2 MagmaHomomorphismByFunctionNC</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X79D0216E871B7051">32.8-3 NaturalHomomorphismByGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X806F892C862F29F9">32.9 <span class="Heading">Mappings that Respect Multiplication</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7BEFF95883EAEC78">32.9-1 RespectsMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7EE4DA097AE9CBC1">32.9-2 RespectsOne</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7F27AE9C84A4DF90">32.9-3 RespectsInverses</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X819DD174829BF3AE">32.9-4 IsGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X81A5A5CF846E5FBF">32.9-5 KernelOfMultiplicativeGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7F09B6E28080DCB4">32.9-6 CoKernelOfMultiplicativeGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X8455A5A67C35178B">32.10 <span class="Heading">Mappings that Respect Addition</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7A3321E878925C3A">32.10-1 RespectsAddition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X8130D8907B92F746">32.10-2 RespectsAdditiveInverses</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7D342736781EB280">32.10-3 RespectsZero</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7B99EF287A8A0BD9">32.10-4 IsAdditiveGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7EC0E9907D6631D6">32.10-5 KernelOfAdditiveGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X813C6D7980213F41">32.10-6 CoKernelOfAdditiveGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X7C24431C81532575">32.11 <span class="Heading">Linear Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X87842ED97FA19973">32.11-1 RespectsScalarMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X780BE6307A3271A9">32.11-2 IsLeftModuleGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7F6841107E59107F">32.11-3 IsLinearMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X7E88C32A82E942DA">32.12 <span class="Heading">Ring Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7C8DA031799B79D5">32.12-1 IsRingGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7988102883675606">32.12-2 IsRingWithOneGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X86B14F908601DEA9">32.12-3 IsAlgebraGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X842AD44679C5BDC2">32.12-4 IsAlgebraWithOneGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X8324DA78879DF4D7">32.12-5 IsFieldHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X7E4A55567BED0F88">32.13 <span class="Heading">General Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X8656AB8A7D672CAE">32.13-1 IsGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X791690817E23D90C">32.13-2 IsConstantTimeAccessGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X81CFF5F87BBEA8AD">32.13-3 IsEndoGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap32.html#X7D6F78587C00CDD0">32.14 <span class="Heading">Technical Matters Concerning General Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7D28581F82481163">32.14-1 IsSPGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X80D02AD183E01F16">32.14-2 IsGeneralMappingFamily</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X86CFADBA7F2FE446">32.14-3 FamilyRange</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7C3736E281A9E505">32.14-4 FamilySource</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7AE54FB67E2E6374">32.14-5 FamiliesOfGeneralMappingsAndRanges</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7E1E26E37C413F6F">32.14-6 GeneralMappingsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap32.html#X7CF92CC37A6BBDA5">32.14-7 TypeOfDefaultGeneralMapping</a></span>
</div></div>
</div>

<h3>32 <span class="Heading">Mappings</span></h3>

<p>A <em>mapping</em> in <strong class="pkg">GAP</strong> is what is called a "function" in mathematics. <strong class="pkg">GAP</strong> also implements <em>generalized mappings</em> in which one element might have several images, these can be imagined as subsets of the cartesian product and are often called "relations".</p>

<p>Most operations are declared for general mappings and therefore this manual often refers to "(general) mappings", unless you deliberately need the generalization you can ignore the "general" bit and just read it as "mappings".</p>

<p>A <em>general mapping</em> <span class="SimpleMath">F</span> in <strong class="pkg">GAP</strong> is described by its source <span class="SimpleMath">S</span>, its range <span class="SimpleMath">R</span>, and a subset <span class="SimpleMath">Rel</span> of the direct product <span class="SimpleMath">S × R</span>, which is called the underlying relation of <span class="SimpleMath">F</span>. <span class="SimpleMath">S</span>, <span class="SimpleMath">R</span>, and <span class="SimpleMath">Rel</span> are generalized domains (see <a href="chap12.html#X7BAF69417BB925F6"><span class="RefLink">12.4</span></a>). The corresponding attributes for general mappings are <code class="func">Source</code> (<a href="chap32.html#X7DE8173F80E07AB1"><span class="RefLink">32.3-8</span></a>), <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>), and <code class="func">UnderlyingRelation</code> (<a href="chap32.html#X784F871383FB599B"><span class="RefLink">32.3-9</span></a>).</p>

<p>Note that general mappings themselves are <em>not</em> 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 <a href="chap32.html#X7E2E16277940FA0B"><span class="RefLink">32.6</span></a>); both should not apply to domains.</p>

<p>Each element of an underlying relation of a general mapping lies in the category of direct product elements (see <code class="func">IsDirectProductElement</code> (<a href="chap32.html#X87FD9FE787023FF0"><span class="RefLink">32.1-1</span></a>)).</p>

<p>For each <span class="SimpleMath">s ∈ S</span>, the set <span class="SimpleMath">{ r ∈ R | (s,r) ∈ Rel }</span> is called the set of <em>images</em> of <span class="SimpleMath">s</span>. Analogously, for <span class="SimpleMath">r ∈ R</span>, the set <span class="SimpleMath">{ s ∈ S | (s,r) ∈ Rel }</span> is called the set of <em>preimages</em> of <span class="SimpleMath">r</span>.</p>

<p>The <em>ordering</em> of general mappings via <code class="code">&lt;</code> is defined by the ordering of source, range, and underlying relation. Specifically, if the source and range domains of <var class="Arg">map1</var> and <var class="Arg">map2</var> are the same, then one considers the union of the preimages of <var class="Arg">map1</var> and <var class="Arg">map2</var> as a strictly ordered set. The smaller of <var class="Arg">map1</var> and <var class="Arg">map2</var> is the one whose image is smaller on the first point of this sequence where they differ.</p>

<p>For mappings which preserve an algebraic structure a <em>kernel</em> is defined. Depending on the structure preserved the operation to compute this kernel is called differently, see Section <a href="chap32.html#X834E02BB7D4B4AE5"><span class="RefLink">32.7</span></a>.</p>

<p>Some technical details of general mappings are described in section <a href="chap32.html#X7E4A55567BED0F88"><span class="RefLink">32.13</span></a>.</p>

<p><a id="X783BAB2683EEA0CC" name="X783BAB2683EEA0CC"></a></p>

<h4>32.1 <span class="Heading">IsDirectProductElement (Filter)</span></h4>

<p><a id="X87FD9FE787023FF0" name="X87FD9FE787023FF0"></a></p>

<h5>32.1-1 IsDirectProductElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsDirectProductElement</code>( <var class="Arg">obj</var> )</td><td class="tdright">(&nbsp;category&nbsp;)</td></tr></table></div>
<p><code class="func">IsDirectProductElement</code> is a subcategory of the meet of <code class="func">IsDenseList</code> (<a href="chap21.html#X870AA9D8798C93DD"><span class="RefLink">21.1-2</span></a>), <code class="func">IsMultiplicativeElementWithInverse</code> (<a href="chap31.html#X7FDB14E57814FA3B"><span class="RefLink">31.14-13</span></a>), <code class="func">IsAdditiveElementWithInverse</code> (<a href="chap31.html#X7C0E4AE883947778"><span class="RefLink">31.14-7</span></a>), and <code class="func">IsCopyable</code> (<a href="chap12.html#X811EFD727EBD1ADC"><span class="RefLink">12.6-1</span></a>), where the arithmetic operations (addition, zero, additive inverse, multiplication, powering, one, inverse) are defined componentwise.</p>

<p>Note that each of these operations will cause an error message if its result for at least one component cannot be formed.</p>

<p>For an object in the filter <code class="func">IsDirectProductElement</code>, <code class="func">ShallowCopy</code> (<a href="chap12.html#X846BC7107C352031"><span class="RefLink">12.7-1</span></a>) returns a mutable plain list with the same entries. The sum and the product of a direct product element and a list in <code class="func">IsListDefault</code> (<a href="chap21.html#X7BAD12E67BFC90DE"><span class="RefLink">21.12-3</span></a>) is the list of sums and products, respectively. The sum and the product of a direct product element and a non-list is the direct product element of componentwise sums and products, respectively.</p>

<p><a id="X7CF6FEFB8290D5CB" name="X7CF6FEFB8290D5CB"></a></p>

<h4>32.2 <span class="Heading">Creating Mappings</span></h4>

<p><a id="X79D0D2F07A14D039" name="X79D0D2F07A14D039"></a></p>

<h5>32.2-1 GeneralMappingByElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneralMappingByElements</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">elms</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>is the general mapping with source <var class="Arg">S</var> and range <var class="Arg">R</var>, and with underlying relation consisting of the collection <var class="Arg">elms</var> of direct product elements.</p>

<p><a id="X7D55E1977ED70E01" name="X7D55E1977ED70E01"></a></p>

<h5>32.2-2 <span class="Heading">MappingByFunction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MappingByFunction</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">fun</var>[, <var class="Arg">invfun</var>] )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MappingByFunction</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">fun</var>, <var class="Arg">false</var>, <var class="Arg">prefun</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p><code class="func">MappingByFunction</code> returns a mapping <code class="code">map</code> with source <var class="Arg">S</var> and range <var class="Arg">R</var>, such that each element <span class="SimpleMath">s</span> of <var class="Arg">S</var> is mapped to the element <var class="Arg">fun</var><span class="SimpleMath">( s )</span>, where <var class="Arg">fun</var> is a <strong class="pkg">GAP</strong> function.</p>

<p>If the argument <var class="Arg">invfun</var> is bound then <code class="code">map</code> is a bijection between <var class="Arg">S</var> and <var class="Arg">R</var>, and the preimage of each element <span class="SimpleMath">r</span> of <var class="Arg">R</var> is given by <var class="Arg">invfun</var><span class="SimpleMath">( r )</span>, where <var class="Arg">invfun</var> is a <strong class="pkg">GAP</strong> function.</p>

<p>If five arguments are given and the fourth argument is <code class="keyw">false</code> then the <strong class="pkg">GAP</strong> function <var class="Arg">prefun</var> can be used to compute a single preimage also if <code class="code">map</code> is not bijective.</p>

<p>The mapping returned by <code class="func">MappingByFunction</code> lies in the filter <code class="func">IsNonSPGeneralMapping</code> (<a href="chap32.html#X7D28581F82481163"><span class="RefLink">32.14-1</span></a>), see <a href="chap32.html#X7D6F78587C00CDD0"><span class="RefLink">32.14</span></a>.</p>

<p><a id="X865FC25A87D36F3D" name="X865FC25A87D36F3D"></a></p>

<h5>32.2-3 InverseGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InverseGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>The <em>inverse general mapping</em> of a general mapping <var class="Arg">map</var> is the general mapping whose underlying relation (see <code class="func">UnderlyingRelation</code> (<a href="chap32.html#X784F871383FB599B"><span class="RefLink">32.3-9</span></a>)) contains a pair <span class="SimpleMath">(r,s)</span> if and only if the underlying relation of <var class="Arg">map</var> contains the pair <span class="SimpleMath">(s,r)</span>.</p>

<p>See the introduction to Chapter <a href="chap32.html#X7C9734B880042C73"><span class="RefLink">32</span></a> for the subtleties concerning the difference between <code class="func">InverseGeneralMapping</code> and <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>).</p>

<p>Note that the inverse general mapping of a mapping <var class="Arg">map</var> is in general only a general mapping. If <var class="Arg">map</var> knows to be bijective its inverse general mapping will know to be a mapping. In this case also <code class="code">Inverse( <var class="Arg">map</var> )</code> works.</p>

<p><a id="X7ED1E4E27CCE2DCA" name="X7ED1E4E27CCE2DCA"></a></p>

<h5>32.2-4 CompositionMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CompositionMapping</code>( <var class="Arg">map1</var>, <var class="Arg">map2</var>, <var class="Arg">...</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p><code class="func">CompositionMapping</code> allows one to compose arbitrarily many general mappings, and delegates each step to <code class="func">CompositionMapping2</code> (<a href="chap32.html#X86486B687B7077AC"><span class="RefLink">32.2-5</span></a>).</p>

<p>Additionally, the properties <code class="func">IsInjective</code> (<a href="chap32.html#X7F065FD7822C0A12"><span class="RefLink">32.3-4</span></a>) and <code class="func">IsSingleValued</code> (<a href="chap32.html#X86D44C8A78BF1981"><span class="RefLink">32.3-2</span></a>) are maintained; if the source of the <span class="SimpleMath">i+1</span>-th general mapping is identical to the range of the <span class="SimpleMath">i</span>-th general mapping, also <code class="func">IsTotal</code> (<a href="chap32.html#X83C7494E828CC9C8"><span class="RefLink">32.3-1</span></a>) and <code class="func">IsSurjective</code> (<a href="chap32.html#X784ECE847E005B8F"><span class="RefLink">32.3-5</span></a>) are maintained. (So one should not call <code class="func">CompositionMapping2</code> (<a href="chap32.html#X86486B687B7077AC"><span class="RefLink">32.2-5</span></a>) directly if one wants to maintain these properties.)</p>

<p>Depending on the types of <var class="Arg">map1</var> and <var class="Arg">map2</var>, 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 <strong class="pkg">GAP</strong> can decompose elements of the source of <var class="Arg">map2</var> into generators) or as an (iterated) composition (see <code class="func">IsCompositionMappingRep</code> (<a href="chap32.html#X7A926D167C3155F6"><span class="RefLink">32.2-6</span></a>)).</p>

<p><a id="X86486B687B7077AC" name="X86486B687B7077AC"></a></p>

<h5>32.2-5 CompositionMapping2</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CompositionMapping2</code>( <var class="Arg">map2</var>, <var class="Arg">map1</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CompositionMapping2General</code>( <var class="Arg">map2</var>, <var class="Arg">map1</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p><code class="func">CompositionMapping2</code> returns the composition of <var class="Arg">map2</var> and <var class="Arg">map1</var>, this is the general mapping that maps an element first under <var class="Arg">map1</var>, and then maps the images under <var class="Arg">map2</var>.</p>

<p>(Note the reverse ordering of arguments in the composition via the multiplication <code class="func">\*</code> (<a href="chap31.html#X8481C9B97B214C23"><span class="RefLink">31.12-1</span></a>).</p>

<p><code class="func">CompositionMapping2General</code> is the method that forms a composite mapping with two constituent mappings. (This is used in some algorithms.)</p>

<p><a id="X7A926D167C3155F6" name="X7A926D167C3155F6"></a></p>

<h5>32.2-6 IsCompositionMappingRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsCompositionMappingRep</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;representation&nbsp;)</td></tr></table></div>
<p>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 <code class="func">ConstituentsCompositionMapping</code> (<a href="chap32.html#X87775B438008DCA5"><span class="RefLink">32.2-7</span></a>).</p>

<p><a id="X87775B438008DCA5" name="X87775B438008DCA5"></a></p>

<h5>32.2-7 ConstituentsCompositionMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConstituentsCompositionMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">map</var> is stored in the representation <code class="func">IsCompositionMappingRep</code> (<a href="chap32.html#X7A926D167C3155F6"><span class="RefLink">32.2-6</span></a>) as composition of two mappings <var class="Arg">map1</var> and <var class="Arg">map2</var>, this function returns the two constituent mappings in a list <code class="code">[ <var class="Arg">map1</var>, <var class="Arg">map2</var> ]</code>.</p>

<p><a id="X795FF8DC785F110A" name="X795FF8DC785F110A"></a></p>

<h5>32.2-8 ZeroMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ZeroMapping</code>( <var class="Arg">S</var>, <var class="Arg">R</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>A zero mapping is a total general mapping that maps each element of its source to the zero element of its range.</p>

<p>(Each mapping with empty source is a zero mapping.)</p>

<p><a id="X7EBAE0368470A603" name="X7EBAE0368470A603"></a></p>

<h5>32.2-9 IdentityMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IdentityMapping</code>( <var class="Arg">D</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>is the bijective mapping with source and range equal to the collection <var class="Arg">D</var>, which maps each element of <var class="Arg">D</var> to itself.</p>

<p><a id="X86452F8587CBAEA0" name="X86452F8587CBAEA0"></a></p>

<h5>32.2-10 <span class="Heading">Embedding</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Embedding</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Embedding</code>( <var class="Arg">S</var>, <var class="Arg">i</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>returns the embedding of the domain <var class="Arg">S</var> in the domain <var class="Arg">T</var>, or in the second form, some domain indexed by the positive integer <var class="Arg">i</var>. The precise natures of the various methods are described elsewhere: for Lie algebras, see <code class="func">LieFamily</code> (<a href="chap64.html#X8725993C7BF386EE"><span class="RefLink">64.1-3</span></a>); for group products, see <a href="chap49.html#X798FDA1386A0EAC6"><span class="RefLink">49.6</span></a> for a general description, or for examples see <a href="chap49.html#X7D39232A84CD8DBD"><span class="RefLink">49.1</span></a> for direct products, <a href="chap49.html#X87FE512E7DB7346C"><span class="RefLink">49.2</span></a> for semidirect products, or <a href="chap49.html#X7DF2AEBC8518FFA4"><span class="RefLink">49.4</span></a> for wreath products; or for magma rings see <a href="chap65.html#X80366F1480ACD8DF"><span class="RefLink">65.3</span></a>.</p>

<p><a id="X8769E8DA80BC96C1" name="X8769E8DA80BC96C1"></a></p>

<h5>32.2-11 <span class="Heading">Projection</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Projection</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Projection</code>( <var class="Arg">S</var>, <var class="Arg">i</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Projection</code>( <var class="Arg">S</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>returns the projection of the domain <var class="Arg">S</var> onto the domain <var class="Arg">T</var>, or in the second form, some domain indexed by the positive integer <var class="Arg">i</var>, or in the third form some natural quotient domain of <var class="Arg">S</var>. Various methods are defined for group products; see <a href="chap49.html#X798FDA1386A0EAC6"><span class="RefLink">49.6</span></a> for a general description, or for examples see <a href="chap49.html#X7D39232A84CD8DBD"><span class="RefLink">49.1</span></a> for direct products, <a href="chap49.html#X87FE512E7DB7346C"><span class="RefLink">49.2</span></a> for semidirect products, <a href="chap49.html#X815AFC537B215D7B"><span class="RefLink">49.3</span></a> for subdirect products, or <a href="chap49.html#X7DF2AEBC8518FFA4"><span class="RefLink">49.4</span></a> for wreath products.</p>

<p><a id="X800014D683A81009" name="X800014D683A81009"></a></p>

<h5>32.2-12 RestrictedMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RestrictedMapping</code>( <var class="Arg">map</var>, <var class="Arg">subdom</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">subdom</var> is a subdomain of the source of the general mapping <var class="Arg">map</var>, this operation returns the restriction of <var class="Arg">map</var> to <var class="Arg">subdom</var>.</p>

<p><a id="X7E5A430D7F838F1C" name="X7E5A430D7F838F1C"></a></p>

<h4>32.3 <span class="Heading">Properties and Attributes of (General) Mappings</span></h4>

<p><a id="X83C7494E828CC9C8" name="X83C7494E828CC9C8"></a></p>

<h5>32.3-1 IsTotal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsTotal</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>is <code class="keyw">true</code> if each element in the source <span class="SimpleMath">S</span> of the general mapping <var class="Arg">map</var> has images, i.e., <span class="SimpleMath">s^<var class="Arg">map</var> ≠ ∅</span> for all <span class="SimpleMath">s ∈ S</span>, and <code class="keyw">false</code> otherwise.</p>

<p><a id="X86D44C8A78BF1981" name="X86D44C8A78BF1981"></a></p>

<h5>32.3-2 IsSingleValued</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSingleValued</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>is <code class="keyw">true</code> if each element in the source <span class="SimpleMath">S</span> of the general mapping <var class="Arg">map</var> has at most one image, i.e., <span class="SimpleMath">|s^<var class="Arg">map</var>| ≤ 1</span> for all <span class="SimpleMath">s ∈ S</span>, and <code class="keyw">false</code> otherwise.</p>

<p>Equivalently, <code class="code">IsSingleValued( <var class="Arg">map</var> )</code> is <code class="keyw">true</code> if and only if the preimages of different elements in <span class="SimpleMath">R</span> are disjoint.</p>

<p><a id="X7CC95EB282854385" name="X7CC95EB282854385"></a></p>

<h5>32.3-3 IsMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p>A <em>mapping</em> <var class="Arg">map</var> is a general mapping that assigns to each element <code class="code">elm</code> of its source a unique element <code class="code">Image( <var class="Arg">map</var>, elm )</code> of its range.</p>

<p>Equivalently, the general mapping <var class="Arg">map</var> is a mapping if and only if it is total and single-valued (see <code class="func">IsTotal</code> (<a href="chap32.html#X83C7494E828CC9C8"><span class="RefLink">32.3-1</span></a>), <code class="func">IsSingleValued</code> (<a href="chap32.html#X86D44C8A78BF1981"><span class="RefLink">32.3-2</span></a>)).</p>

<p><a id="X7F065FD7822C0A12" name="X7F065FD7822C0A12"></a></p>

<h5>32.3-4 IsInjective</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsInjective</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>is <code class="keyw">true</code> if the images of different elements in the source <span class="SimpleMath">S</span> of the general mapping <var class="Arg">map</var> are disjoint, i.e., <span class="SimpleMath">x^<var class="Arg">map</var> ∩ y^<var class="Arg">map</var> = ∅</span> for <span class="SimpleMath">x ≠ y ∈ S</span>, and <code class="keyw">false</code> otherwise.</p>

<p>Equivalently, <code class="code">IsInjective( <var class="Arg">map</var> )</code> is <code class="keyw">true</code> if and only if each element in the range of <var class="Arg">map</var> has at most one preimage in <span class="SimpleMath">S</span>.</p>

<p><a id="X784ECE847E005B8F" name="X784ECE847E005B8F"></a></p>

<h5>32.3-5 IsSurjective</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSurjective</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>is <code class="keyw">true</code> if each element in the range <span class="SimpleMath">R</span> of the general mapping <var class="Arg">map</var> has preimages in the source <span class="SimpleMath">S</span> of <var class="Arg">map</var>, i.e., <span class="SimpleMath">{ s ∈ S ∣ x ∈ s^<var class="Arg">map</var> } ≠ ∅</span> for all <span class="SimpleMath">x ∈ R</span>, and <code class="keyw">false</code> otherwise.</p>

<p><a id="X878F56AB7B342767" name="X878F56AB7B342767"></a></p>

<h5>32.3-6 IsBijective</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsBijective</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>A general mapping <var class="Arg">map</var> is <em>bijective</em> if and only if it is an injective and surjective mapping (see <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>), <code class="func">IsInjective</code> (<a href="chap32.html#X7F065FD7822C0A12"><span class="RefLink">32.3-4</span></a>), <code class="func">IsSurjective</code> (<a href="chap32.html#X784ECE847E005B8F"><span class="RefLink">32.3-5</span></a>)).</p>

<p><a id="X7B6FD7277CDE9FCB" name="X7B6FD7277CDE9FCB"></a></p>

<h5>32.3-7 Range</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Range</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>The range of a general mapping.</p>

<p><a id="X7DE8173F80E07AB1" name="X7DE8173F80E07AB1"></a></p>

<h5>32.3-8 Source</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Source</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>The source of a general mapping.</p>

<p><a id="X784F871383FB599B" name="X784F871383FB599B"></a></p>

<h5>32.3-9 UnderlyingRelation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; UnderlyingRelation</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>The <em>underlying relation</em> of a general mapping <var class="Arg">map</var> is the domain of pairs <span class="SimpleMath">(s,r)</span>, with <span class="SimpleMath">s</span> in the source and <span class="SimpleMath">r</span> in the range of <var class="Arg">map</var> (see <code class="func">Source</code> (<a href="chap32.html#X7DE8173F80E07AB1"><span class="RefLink">32.3-8</span></a>), <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>)), and <span class="SimpleMath">r ∈</span> <code class="code">ImagesElm( <var class="Arg">map</var>, </code><span class="SimpleMath">s</span><code class="code"> )</code>.</p>

<p>Each element of the underlying relation is represented by a direct product element (see <code class="func">IsDirectProductElement</code> (<a href="chap32.html#X87FD9FE787023FF0"><span class="RefLink">32.1-1</span></a>)).</p>

<p><a id="X786581DE871A47D0" name="X786581DE871A47D0"></a></p>

<h5>32.3-10 UnderlyingGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; UnderlyingGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>attribute for underlying relations of general mappings</p>

<p><a id="X83B4FF15847F06FC" name="X83B4FF15847F06FC"></a></p>

<h4>32.4 <span class="Heading">Images under Mappings</span></h4>

<p><a id="X7D23C1CE863DACD8" name="X7D23C1CE863DACD8"></a></p>

<h5>32.4-1 ImagesSource</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ImagesSource</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>is the set of images of the source of the general mapping <var class="Arg">map</var>.</p>

<p><code class="func">ImagesSource</code> delegates to <code class="func">ImagesSet</code> (<a href="chap32.html#X8781348F7F5796A0"><span class="RefLink">32.4-4</span></a>), it is introduced only to store the image of <var class="Arg">map</var> as attribute value.</p>

<p><a id="X85ADB89B7C8DD7D0" name="X85ADB89B7C8DD7D0"></a></p>

<h5>32.4-2 ImagesRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ImagesRepresentative</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elm</var> is an element of the source of the general mapping <var class="Arg">map</var> then <code class="func">ImagesRepresentative</code> returns either a representative of the set of images of <var class="Arg">elm</var> under <var class="Arg">map</var> or <code class="keyw">fail</code>, the latter if and only if <var class="Arg">elm</var> has no images under <var class="Arg">map</var>.</p>

<p>Anything may happen if <var class="Arg">elm</var> is not an element of the source of <var class="Arg">map</var>.</p>

<p><a id="X7D51184B7EE5B2CF" name="X7D51184B7EE5B2CF"></a></p>

<h5>32.4-3 ImagesElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ImagesElm</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elm</var> is an element of the source of the general mapping <var class="Arg">map</var> then <code class="func">ImagesElm</code> returns the set of all images of <var class="Arg">elm</var> under <var class="Arg">map</var>.</p>

<p>Anything may happen if <var class="Arg">elm</var> is not an element of the source of <var class="Arg">map</var>.</p>

<p><a id="X8781348F7F5796A0" name="X8781348F7F5796A0"></a></p>

<h5>32.4-4 ImagesSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ImagesSet</code>( <var class="Arg">map</var>, <var class="Arg">elms</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elms</var> is a subset of the source of the general mapping <var class="Arg">map</var> then <code class="func">ImagesSet</code> returns the set of all images of <var class="Arg">elms</var> under <var class="Arg">map</var>.</p>

<p>The result will be either a proper set or a domain. Anything may happen if <var class="Arg">elms</var> is not a subset of the source of <var class="Arg">map</var>.</p>

<p><a id="X7CFAB0157BFB1806" name="X7CFAB0157BFB1806"></a></p>

<h5>32.4-5 ImageElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ImageElm</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elm</var> is an element of the source of the total and single-valued mapping <var class="Arg">map</var> then <code class="func">ImageElm</code> returns the unique image of <var class="Arg">elm</var> under <var class="Arg">map</var>.</p>

<p>Anything may happen if <var class="Arg">elm</var> is not an element of the source of <var class="Arg">map</var>.</p>

<p><a id="X87F4D35A826599C6" name="X87F4D35A826599C6"></a></p>

<h5>32.4-6 <span class="Heading">Image</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Image</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Image</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Image</code>( <var class="Arg">map</var>, <var class="Arg">coll</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p><code class="code">Image( <var class="Arg">map</var> )</code> is the <em>image</em> of the general mapping <var class="Arg">map</var>, i.e., the subset of elements of the range of <var class="Arg">map</var> that are actually values of <var class="Arg">map</var>. <em>Note</em> that in this case the argument may also be multi-valued.</p>

<p><code class="code">Image( <var class="Arg">map</var>, <var class="Arg">elm</var> )</code> is the image of the element <var class="Arg">elm</var> of the source of the mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the unique element of the range to which <var class="Arg">map</var> maps <var class="Arg">elm</var>. This can also be expressed as <var class="Arg">elm</var><code class="code">^</code><var class="Arg">map</var>. Note that <var class="Arg">map</var> must be total and single valued, a multi valued general mapping is not allowed (see <code class="func">Images</code> (<a href="chap32.html#X86114B2E7E77488C"><span class="RefLink">32.4-7</span></a>)).</p>

<p><code class="code">Image( <var class="Arg">map</var>, <var class="Arg">coll</var> )</code> is the image of the subset <var class="Arg">coll</var> of the source of the mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the subset of the range to which <var class="Arg">map</var> maps elements of <var class="Arg">coll</var>. <var class="Arg">coll</var> may be a proper set or a domain. The result will be either a proper set or a domain. Note that in this case <var class="Arg">map</var> may also be multi-valued. (If <var class="Arg">coll</var> and the result are lists then the positions of entries do in general <em>not</em> correspond.)</p>

<p><code class="func">Image</code> delegates to <code class="func">ImagesSource</code> (<a href="chap32.html#X7D23C1CE863DACD8"><span class="RefLink">32.4-1</span></a>) when called with one argument, and to <code class="func">ImageElm</code> (<a href="chap32.html#X7CFAB0157BFB1806"><span class="RefLink">32.4-5</span></a>) resp. <code class="func">ImagesSet</code> (<a href="chap32.html#X8781348F7F5796A0"><span class="RefLink">32.4-4</span></a>) when called with two arguments.</p>

<p>If the second argument is not an element or a subset of the source of the first argument, an error is signalled.</p>

<p><a id="X86114B2E7E77488C" name="X86114B2E7E77488C"></a></p>

<h5>32.4-7 <span class="Heading">Images</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Images</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Images</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Images</code>( <var class="Arg">map</var>, <var class="Arg">coll</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p><code class="code">Images( <var class="Arg">map</var> )</code> is the <em>image</em> of the general mapping <var class="Arg">map</var>, i.e., the subset of elements of the range of <var class="Arg">map</var> that are actually values of <var class="Arg">map</var>.</p>

<p><code class="code">Images( <var class="Arg">map</var>, <var class="Arg">elm</var> )</code> is the set of images of the element <var class="Arg">elm</var> of the source of the general mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the set of elements of the range to which <var class="Arg">map</var> maps <var class="Arg">elm</var>.</p>

<p><code class="code">Images( <var class="Arg">map</var>, <var class="Arg">coll</var> )</code> is the set of images of the subset <var class="Arg">coll</var> of the source of the general mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the subset of the range to which <var class="Arg">map</var> maps elements of <var class="Arg">coll</var>. <var class="Arg">coll</var> may be a proper set or a domain. The result will be either a proper set or a domain. (If <var class="Arg">coll</var> and the result are lists then the positions of entries do in general <em>not</em> correspond.)</p>

<p><code class="func">Images</code> delegates to <code class="func">ImagesSource</code> (<a href="chap32.html#X7D23C1CE863DACD8"><span class="RefLink">32.4-1</span></a>) when called with one argument, and to <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) resp. <code class="func">ImagesSet</code> (<a href="chap32.html#X8781348F7F5796A0"><span class="RefLink">32.4-4</span></a>) when called with two arguments.</p>

<p>If the second argument is not an element or a subset of the source of the first argument, an error is signalled.</p>

<p><a id="X79BB1EC07C828667" name="X79BB1EC07C828667"></a></p>

<h4>32.5 <span class="Heading">Preimages under Mappings</span></h4>

<p><a id="X78EF1FE77B0973C0" name="X78EF1FE77B0973C0"></a></p>

<h5>32.5-1 PreImagesRange</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImagesRange</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>is the set of preimages of the range of the general mapping <var class="Arg">map</var>.</p>

<p><code class="func">PreImagesRange</code> delegates to <code class="func">PreImagesSet</code> (<a href="chap32.html#X856BAFC87B2D2811"><span class="RefLink">32.5-5</span></a>), it is introduced only to store the preimage of <var class="Arg">map</var> as attribute value.</p>

<p><a id="X7FBB830C8729E995" name="X7FBB830C8729E995"></a></p>

<h5>32.5-2 PreImagesElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImagesElm</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elm</var> is an element of the range of the general mapping <var class="Arg">map</var> then <code class="func">PreImagesElm</code> returns the set of all preimages of <var class="Arg">elm</var> under <var class="Arg">map</var>.</p>

<p>Anything may happen if <var class="Arg">elm</var> is not an element of the range of <var class="Arg">map</var>.</p>

<p><a id="X7D212F727CAE971A" name="X7D212F727CAE971A"></a></p>

<h5>32.5-3 PreImageElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImageElm</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elm</var> is an element of the range of the injective and surjective general mapping <var class="Arg">map</var> then <code class="func">PreImageElm</code> returns the unique preimage of <var class="Arg">elm</var> under <var class="Arg">map</var>.</p>

<p>Anything may happen if <var class="Arg">elm</var> is not an element of the range of <var class="Arg">map</var>.</p>

<p><a id="X7AE24A1586B7DE79" name="X7AE24A1586B7DE79"></a></p>

<h5>32.5-4 PreImagesRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImagesRepresentative</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elm</var> is an element of the range of the general mapping <var class="Arg">map</var> then <code class="func">PreImagesRepresentative</code> returns either a representative of the set of preimages of <var class="Arg">elm</var> under <var class="Arg">map</var> or <code class="keyw">fail</code>, the latter if and only if <var class="Arg">elm</var> has no preimages under <var class="Arg">map</var>.</p>

<p>Anything may happen if <var class="Arg">elm</var> is not an element of the range of <var class="Arg">map</var>.</p>

<p><a id="X856BAFC87B2D2811" name="X856BAFC87B2D2811"></a></p>

<h5>32.5-5 PreImagesSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImagesSet</code>( <var class="Arg">map</var>, <var class="Arg">elms</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>If <var class="Arg">elms</var> is a subset of the range of the general mapping <var class="Arg">map</var> then <code class="func">PreImagesSet</code> returns the set of all preimages of <var class="Arg">elms</var> under <var class="Arg">map</var>.</p>

<p>Anything may happen if <var class="Arg">elms</var> is not a subset of the range of <var class="Arg">map</var>.</p>

<p><a id="X836FAEAC78B55BF4" name="X836FAEAC78B55BF4"></a></p>

<h5>32.5-6 <span class="Heading">PreImage</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImage</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImage</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImage</code>( <var class="Arg">map</var>, <var class="Arg">coll</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p><code class="code">PreImage( <var class="Arg">map</var> )</code> is the preimage of the general mapping <var class="Arg">map</var>, i.e., the subset of elements of the source of <var class="Arg">map</var> that actually have values under <var class="Arg">map</var>. Note that in this case the argument may also be non-injective or non-surjective.</p>

<p><code class="code">PreImage( <var class="Arg">map</var>, <var class="Arg">elm</var> )</code> is the preimage of the element <var class="Arg">elm</var> of the range of the injective and surjective mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the unique element of the source which is mapped under <var class="Arg">map</var> to <var class="Arg">elm</var>. Note that <var class="Arg">map</var> must be injective and surjective (see <code class="func">PreImages</code> (<a href="chap32.html#X85C8590E832002EF"><span class="RefLink">32.5-7</span></a>)).</p>

<p><code class="code">PreImage( <var class="Arg">map</var>, <var class="Arg">coll</var> )</code> is the preimage of the subset <var class="Arg">coll</var> of the range of the general mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the subset of the source which is mapped under <var class="Arg">map</var> to elements of <var class="Arg">coll</var>. <var class="Arg">coll</var> may be a proper set or a domain. The result will be either a proper set or a domain. Note that in this case <var class="Arg">map</var> may also be non-injective or non-surjective. (If <var class="Arg">coll</var> and the result are lists then the positions of entries do in general <em>not</em> correspond.)</p>

<p><code class="func">PreImage</code> delegates to <code class="func">PreImagesRange</code> (<a href="chap32.html#X78EF1FE77B0973C0"><span class="RefLink">32.5-1</span></a>) when called with one argument, and to <code class="func">PreImageElm</code> (<a href="chap32.html#X7D212F727CAE971A"><span class="RefLink">32.5-3</span></a>) resp. <code class="func">PreImagesSet</code> (<a href="chap32.html#X856BAFC87B2D2811"><span class="RefLink">32.5-5</span></a>) when called with two arguments.</p>

<p>If the second argument is not an element or a subset of the range of the first argument, an error is signalled.</p>

<p><a id="X85C8590E832002EF" name="X85C8590E832002EF"></a></p>

<h5>32.5-7 <span class="Heading">PreImages</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImages</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImages</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PreImages</code>( <var class="Arg">map</var>, <var class="Arg">coll</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p><code class="code">PreImages( <var class="Arg">map</var> )</code> is the preimage of the general mapping <var class="Arg">map</var>, i.e., the subset of elements of the source of <var class="Arg">map</var> that have actually values under <var class="Arg">map</var>.</p>

<p><code class="code">PreImages( <var class="Arg">map</var>, <var class="Arg">elm</var> )</code> is the set of preimages of the element <var class="Arg">elm</var> of the range of the general mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the set of elements of the source which <var class="Arg">map</var> maps to <var class="Arg">elm</var>.</p>

<p><code class="code">PreImages( <var class="Arg">map</var>, <var class="Arg">coll</var> )</code> is the set of images of the subset <var class="Arg">coll</var> of the range of the general mapping <var class="Arg">map</var> under <var class="Arg">map</var>, i.e., the subset of the source which <var class="Arg">map</var> maps to elements of <var class="Arg">coll</var>. <var class="Arg">coll</var> may be a proper set or a domain. The result will be either a proper set or a domain. (If <var class="Arg">coll</var> and the result are lists then the positions of entries do in general <em>not</em> correspond.)</p>

<p><code class="func">PreImages</code> delegates to <code class="func">PreImagesRange</code> (<a href="chap32.html#X78EF1FE77B0973C0"><span class="RefLink">32.5-1</span></a>) when called with one argument, and to <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>) resp. <code class="func">PreImagesSet</code> (<a href="chap32.html#X856BAFC87B2D2811"><span class="RefLink">32.5-5</span></a>) when called with two arguments.</p>

<p>If the second argument is not an element or a subset of the range of the first argument, an error is signalled.</p>

<p><a id="X7E2E16277940FA0B" name="X7E2E16277940FA0B"></a></p>

<h4>32.6 <span class="Heading">Arithmetic Operations for General Mappings</span></h4>

<p>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>

<p>For two general mappings with same source, range, preimage, and image, the <em>sum</em> 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>

<p><em>Scalar multiplication</em> of general mappings is defined likewise.</p>

<p>The <em>product</em> of two general mappings is defined as the composition. This multiplication is always associative. In addition to the composition via <code class="code">*</code>, general mappings can be composed –in reversed order– via <code class="func">CompositionMapping</code> (<a href="chap32.html#X7ED1E4E27CCE2DCA"><span class="RefLink">32.2-4</span></a>).</p>

<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 <code class="func">One</code> (<a href="chap31.html#X8046262384895B2A"><span class="RefLink">31.10-2</span></a>) and <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>) for general mappings as follows. <code class="func">One</code> (<a href="chap31.html#X8046262384895B2A"><span class="RefLink">31.10-2</span></a>) returns <code class="keyw">fail</code> when called for a general mapping whose source and range differ, otherwise <code class="func">One</code> (<a href="chap31.html#X8046262384895B2A"><span class="RefLink">31.10-2</span></a>) returns the identity mapping of the source. (Note that the source may differ from the preimage). <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>) returns <code class="keyw">fail</code> when called for a non-bijective general mapping or for a general mapping whose source and range differ; otherwise <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>) returns the inverse mapping.</p>

<p>Besides the usual inverse of multiplicative elements, which means that <code class="code">Inverse( <var class="Arg">g</var> ) * <var class="Arg">g</var> = <var class="Arg">g</var> * Inverse( <var class="Arg">g</var> ) = One( <var class="Arg">g</var> )</code>, for general mappings we have the attribute <code class="func">InverseGeneralMapping</code> (<a href="chap32.html#X865FC25A87D36F3D"><span class="RefLink">32.2-3</span></a>). If <var class="Arg">F</var> is a general mapping with source <span class="SimpleMath">S</span>, range <span class="SimpleMath">R</span>, and underlying relation <span class="SimpleMath">Rel</span> then <code class="code">InverseGeneralMapping( <var class="Arg">F</var> )</code> has source <span class="SimpleMath">R</span>, range <span class="SimpleMath">S</span>, and underlying relation <span class="SimpleMath">{ (r,s) ∣ (s,r) ∈ Rel }</span>. 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>

<p><code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>) may delegate to <code class="func">InverseGeneralMapping</code> (<a href="chap32.html#X865FC25A87D36F3D"><span class="RefLink">32.2-3</span></a>). <code class="func">InverseGeneralMapping</code> (<a href="chap32.html#X865FC25A87D36F3D"><span class="RefLink">32.2-3</span></a>) must not delegate to <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>), but a known value of <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>) may be fetched. So methods to compute the inverse of a general mapping should be installed for <code class="func">InverseGeneralMapping</code> (<a href="chap32.html#X865FC25A87D36F3D"><span class="RefLink">32.2-3</span></a>).</p>

<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 <strong class="pkg">GAP</strong>.)</p>

<p><a id="X834E02BB7D4B4AE5" name="X834E02BB7D4B4AE5"></a></p>

<h4>32.7 <span class="Heading">Mappings which are Compatible with Algebraic Structures</span></h4>

<p>From an algebraical point of view, the most important mappings are those which are compatible with a structure. For Magmas, Groups and Rings, <strong class="pkg">GAP</strong> supports the following four types of such mappings:</p>

<ol>
<li><p>General mappings that respect multiplication</p>

</li>
<li><p>General mappings that respect addition</p>

</li>
<li><p>General mappings that respect scalar mult.</p>

</li>
<li><p>General mappings that respect multiplicative and additive structure</p>

</li>
</ol>
<p>(Very technical note: <strong class="pkg">GAP</strong> defines categories <code class="code">IsSPGeneralMapping</code> and <code class="code">IsNonSPGeneralMapping</code>. The distinction between these is orthogonal to the structure compatibility described here and should not be confused.)</p>

<p><a id="X8008FCCC7F4C731F" name="X8008FCCC7F4C731F"></a></p>

<h4>32.8 <span class="Heading">Magma Homomorphisms</span></h4>

<p><a id="X7DC72CF28539A251" name="X7DC72CF28539A251"></a></p>

<h5>32.8-1 IsMagmaHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsMagmaHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p>A <em>magma homomorphism</em> is a total single valued mapping which respects multiplication.</p>

<p><a id="X8181676787E760A2" name="X8181676787E760A2"></a></p>

<h5>32.8-2 MagmaHomomorphismByFunctionNC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MagmaHomomorphismByFunctionNC</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">fn</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Creates the homomorphism from <var class="Arg">G</var> to <var class="Arg">H</var> without checking that <var class="Arg">fn</var> is a homomorphism.</p>

<p><a id="X79D0216E871B7051" name="X79D0216E871B7051"></a></p>

<h5>32.8-3 NaturalHomomorphismByGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NaturalHomomorphismByGenerators</code>( <var class="Arg">f</var>, <var class="Arg">s</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>returns a mapping from the magma <var class="Arg">f</var> with <span class="SimpleMath">n</span> generators to the magma <var class="Arg">s</var> with <span class="SimpleMath">n</span> generators, which maps the <span class="SimpleMath">i</span>-th generator of <var class="Arg">f</var> to the <span class="SimpleMath">i</span>-th generator of <var class="Arg">s</var>.</p>

<p><a id="X806F892C862F29F9" name="X806F892C862F29F9"></a></p>

<h4>32.9 <span class="Heading">Mappings that Respect Multiplication</span></h4>

<p><a id="X7BEFF95883EAEC78" name="X7BEFF95883EAEC78"></a></p>

<h5>32.9-1 RespectsMultiplication</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RespectsMultiplication</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping with underlying relation <span class="SimpleMath">F ⊆ S × R</span>, where <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are the source and the range of <var class="Arg">mapp</var>, respectively. Then <code class="func">RespectsMultiplication</code> returns <code class="keyw">true</code> if <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are magmas such that <span class="SimpleMath">(s_1,r_1), (s_2,r_2) ∈ F</span> implies <span class="SimpleMath">(s_1 * s_2,r_1 * r_2) ∈ F</span>, and <code class="keyw">false</code> otherwise.</p>

<p>If <var class="Arg">mapp</var> is single-valued then <code class="func">RespectsMultiplication</code> returns <code class="keyw">true</code> if and only if the equation <code class="code"><var class="Arg">s1</var>^<var class="Arg">mapp</var> * <var class="Arg">s2</var>^<var class="Arg">mapp</var> = (<var class="Arg">s1</var> * <var class="Arg">s2</var>)^<var class="Arg">mapp</var></code> holds for all <var class="Arg">s1</var>, <var class="Arg">s2</var> in <span class="SimpleMath">S</span>.</p>

<p><a id="X7EE4DA097AE9CBC1" name="X7EE4DA097AE9CBC1"></a></p>

<h5>32.9-2 RespectsOne</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RespectsOne</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping with underlying relation <span class="SimpleMath">F ⊆ <var class="Arg">S</var> × <var class="Arg">R</var></span>, where <var class="Arg">S</var> and <var class="Arg">R</var> are the source and the range of <var class="Arg">mapp</var>, respectively. Then <code class="func">RespectsOne</code> returns <code class="keyw">true</code> if <var class="Arg">S</var> and <var class="Arg">R</var> are magmas-with-one such that <span class="SimpleMath">(</span><code class="code">One(<var class="Arg">S</var>)</code><span class="SimpleMath">,</span><code class="code">One(<var class="Arg">R</var>)</code><span class="SimpleMath">) ∈ F</span>, and <code class="keyw">false</code> otherwise.</p>

<p>If <var class="Arg">mapp</var> is single-valued then <code class="func">RespectsOne</code> returns <code class="keyw">true</code> if and only if the equation <code class="code">One( <var class="Arg">S</var> )^<var class="Arg">mapp</var> = One( <var class="Arg">R</var> )</code> holds.</p>

<p><a id="X7F27AE9C84A4DF90" name="X7F27AE9C84A4DF90"></a></p>

<h5>32.9-3 RespectsInverses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RespectsInverses</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping with underlying relation <span class="SimpleMath">F ⊆ <var class="Arg">S</var> × <var class="Arg">R</var></span>, where <var class="Arg">S</var> and <var class="Arg">R</var> are the source and the range of <var class="Arg">mapp</var>, respectively. Then <code class="func">RespectsInverses</code> returns <code class="keyw">true</code> if <var class="Arg">S</var> and <var class="Arg">R</var> are magmas-with-inverses such that, for <span class="SimpleMath">s ∈ <var class="Arg">S</var></span> and <span class="SimpleMath">r ∈ <var class="Arg">R</var></span>, <span class="SimpleMath">(s,r) ∈ F</span> implies <span class="SimpleMath">(s^{-1},r^{-1}) ∈ F</span>, and <code class="keyw">false</code> otherwise.</p>

<p>If <var class="Arg">mapp</var> is single-valued then <code class="func">RespectsInverses</code> returns <code class="keyw">true</code> if and only if the equation <code class="code">Inverse( <var class="Arg">s</var> )^<var class="Arg">mapp</var> = Inverse( <var class="Arg">s</var>^<var class="Arg">mapp</var> )</code> holds for all <var class="Arg">s</var> in <span class="SimpleMath">S</span>.</p>

<p><a id="X819DD174829BF3AE" name="X819DD174829BF3AE"></a></p>

<h5>32.9-4 IsGroupGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsGroupGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsGroupHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p>A <em>group general mapping</em> is a mapping which respects multiplication and inverses. If it is total and single valued it is called a <em>group homomorphism</em>.</p>

<p>Chapter <a href="chap40.html#X83702FC27B3C3098"><span class="RefLink">40</span></a> explains group homomorphisms in more detail.</p>

<p><a id="X81A5A5CF846E5FBF" name="X81A5A5CF846E5FBF"></a></p>

<h5>32.9-5 KernelOfMultiplicativeGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; KernelOfMultiplicativeGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping. Then <code class="func">KernelOfMultiplicativeGeneralMapping</code> returns the set of all elements in the source of <var class="Arg">mapp</var> that have the identity of the range in their set of images.</p>

<p>(This is a monoid if <var class="Arg">mapp</var> respects multiplication and one, and if the source of <var class="Arg">mapp</var> is associative.)</p>

<p><a id="X7F09B6E28080DCB4" name="X7F09B6E28080DCB4"></a></p>

<h5>32.9-6 CoKernelOfMultiplicativeGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CoKernelOfMultiplicativeGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping. Then <code class="func">CoKernelOfMultiplicativeGeneralMapping</code> returns the set of all elements in the range of <var class="Arg">mapp</var> that have the identity of the source in their set of preimages.</p>

<p>(This is a monoid if <var class="Arg">mapp</var> respects multiplication and one, and if the range of <var class="Arg">mapp</var> is associative.)</p>

<p><a id="X8455A5A67C35178B" name="X8455A5A67C35178B"></a></p>

<h4>32.10 <span class="Heading">Mappings that Respect Addition</span></h4>

<p><a id="X7A3321E878925C3A" name="X7A3321E878925C3A"></a></p>

<h5>32.10-1 RespectsAddition</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RespectsAddition</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping with underlying relation <span class="SimpleMath">F ⊆ S × R</span>, where <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are the source and the range of <var class="Arg">mapp</var>, respectively. Then <code class="func">RespectsAddition</code> returns <code class="keyw">true</code> if <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are additive magmas such that <span class="SimpleMath">(s_1,r_1), (s_2,r_2) ∈ F</span> implies <span class="SimpleMath">(s_1 + s_2,r_1 + r_2) ∈ F</span>, and <code class="keyw">false</code> otherwise.</p>

<p>If <var class="Arg">mapp</var> is single-valued then <code class="func">RespectsAddition</code> returns <code class="keyw">true</code> if and only if the equation <code class="code"><var class="Arg">s1</var>^<var class="Arg">mapp</var> + <var class="Arg">s2</var>^<var class="Arg">mapp</var> = (<var class="Arg">s1</var>+<var class="Arg">s2</var>)^<var class="Arg">mapp</var></code> holds for all <var class="Arg">s1</var>, <var class="Arg">s2</var> in <span class="SimpleMath">S</span>.</p>

<p><a id="X8130D8907B92F746" name="X8130D8907B92F746"></a></p>

<h5>32.10-2 RespectsAdditiveInverses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RespectsAdditiveInverses</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping with underlying relation <span class="SimpleMath">F ⊆ S × R</span>, where <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are the source and the range of <var class="Arg">mapp</var>, respectively. Then <code class="func">RespectsAdditiveInverses</code> returns <code class="keyw">true</code> if <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are additive-magmas-with-inverses such that <span class="SimpleMath">(s,r) ∈ F</span> implies <span class="SimpleMath">(-s,-r) ∈ F</span>, and <code class="keyw">false</code> otherwise.</p>

<p>If <var class="Arg">mapp</var> is single-valued then <code class="func">RespectsAdditiveInverses</code> returns <code class="keyw">true</code> if and only if the equation <code class="code">AdditiveInverse( <var class="Arg">s</var> )^<var class="Arg">mapp</var> = AdditiveInverse( <var class="Arg">s</var>^<var class="Arg">mapp</var> )</code> holds for all <var class="Arg">s</var> in <span class="SimpleMath">S</span>.</p>

<p><a id="X7D342736781EB280" name="X7D342736781EB280"></a></p>

<h5>32.10-3 RespectsZero</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RespectsZero</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping with underlying relation <span class="SimpleMath">F ⊆ <var class="Arg">S</var> × <var class="Arg">R</var></span>, where <var class="Arg">S</var> and <var class="Arg">R</var> are the source and the range of <var class="Arg">mapp</var>, respectively. Then <code class="func">RespectsZero</code> returns <code class="keyw">true</code> if <var class="Arg">S</var> and <var class="Arg">R</var> are additive-magmas-with-zero such that <span class="SimpleMath">(</span><code class="code">Zero(<var class="Arg">S</var>)</code><span class="SimpleMath">,</span><code class="code">Zero(<var class="Arg">R</var>)</code><span class="SimpleMath">) ∈ F</span>, and <code class="keyw">false</code> otherwise.</p>

<p>If <var class="Arg">mapp</var> is single-valued then <code class="func">RespectsZero</code> returns <code class="keyw">true</code> if and only if the equation <code class="code">Zero( <var class="Arg">S</var> )^<var class="Arg">mapp</var> = Zero( <var class="Arg">R</var> )</code> holds.</p>

<p><a id="X7B99EF287A8A0BD9" name="X7B99EF287A8A0BD9"></a></p>

<h5>32.10-4 IsAdditiveGroupGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsAdditiveGroupGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsAdditiveGroupHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p><code class="func">IsAdditiveGroupGeneralMapping</code> specifies whether a general mapping <var class="Arg">mapp</var> respects addition (see <code class="func">RespectsAddition</code> (<a href="chap32.html#X7A3321E878925C3A"><span class="RefLink">32.10-1</span></a>)) and respects additive inverses (see <code class="func">RespectsAdditiveInverses</code> (<a href="chap32.html#X8130D8907B92F746"><span class="RefLink">32.10-2</span></a>)).</p>

<p><code class="func">IsAdditiveGroupHomomorphism</code> is a synonym for the meet of <code class="func">IsAdditiveGroupGeneralMapping</code> and <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>).</p>

<p><a id="X7EC0E9907D6631D6" name="X7EC0E9907D6631D6"></a></p>

<h5>32.10-5 KernelOfAdditiveGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; KernelOfAdditiveGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping. Then <code class="func">KernelOfAdditiveGeneralMapping</code> returns the set of all elements in the source of <var class="Arg">mapp</var> that have the zero of the range in their set of images.</p>

<p><a id="X813C6D7980213F41" name="X813C6D7980213F41"></a></p>

<h5>32.10-6 CoKernelOfAdditiveGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CoKernelOfAdditiveGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping. Then <code class="func">CoKernelOfAdditiveGeneralMapping</code> returns the set of all elements in the range of <var class="Arg">mapp</var> that have the zero of the source in their set of preimages.</p>

<p><a id="X7C24431C81532575" name="X7C24431C81532575"></a></p>

<h4>32.11 <span class="Heading">Linear Mappings</span></h4>

<p>Also see Sections <a href="chap32.html#X806F892C862F29F9"><span class="RefLink">32.9</span></a>, <a href="chap32.html#X8455A5A67C35178B"><span class="RefLink">32.10</span></a>, and <code class="func">KernelOfMultiplicativeGeneralMapping</code> (<a href="chap32.html#X81A5A5CF846E5FBF"><span class="RefLink">32.9-5</span></a>), <code class="func">CoKernelOfMultiplicativeGeneralMapping</code> (<a href="chap32.html#X7F09B6E28080DCB4"><span class="RefLink">32.9-6</span></a>).</p>

<p><a id="X87842ED97FA19973" name="X87842ED97FA19973"></a></p>

<h5>32.11-1 RespectsScalarMultiplication</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RespectsScalarMultiplication</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>Let <var class="Arg">mapp</var> be a general mapping, with underlying relation <span class="SimpleMath">F ⊆ S × R</span>, where <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are the source and the range of <var class="Arg">mapp</var>, respectively. Then <code class="func">RespectsScalarMultiplication</code> returns <code class="keyw">true</code> if <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> are left modules with the left acting domain <span class="SimpleMath">D</span> of <span class="SimpleMath">S</span> contained in the left acting domain of <span class="SimpleMath">R</span> and such that <span class="SimpleMath">(s,r) ∈ F</span> implies <span class="SimpleMath">(c * s,c * r) ∈ F</span> for all <span class="SimpleMath">c ∈ D</span>, and <code class="keyw">false</code> otherwise.</p>

<p>If <var class="Arg">mapp</var> is single-valued then <code class="func">RespectsScalarMultiplication</code> returns <code class="keyw">true</code> if and only if the equation <code class="code"><var class="Arg">c</var> * <var class="Arg">s</var>^<var class="Arg">mapp</var> = (<var class="Arg">c</var> * <var class="Arg">s</var>)^<var class="Arg">mapp</var></code> holds for all <var class="Arg">c</var> in <span class="SimpleMath">D</span> and <var class="Arg">s</var> in <span class="SimpleMath">S</span>.</p>

<p><a id="X780BE6307A3271A9" name="X780BE6307A3271A9"></a></p>

<h5>32.11-2 IsLeftModuleGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsLeftModuleGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsLeftModuleHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p><code class="func">IsLeftModuleGeneralMapping</code> specifies whether a general mapping <var class="Arg">mapp</var> satisfies the property <code class="func">IsAdditiveGroupGeneralMapping</code> (<a href="chap32.html#X7B99EF287A8A0BD9"><span class="RefLink">32.10-4</span></a>) and respects scalar multiplication (see <code class="func">RespectsScalarMultiplication</code> (<a href="chap32.html#X87842ED97FA19973"><span class="RefLink">32.11-1</span></a>)).</p>

<p><code class="func">IsLeftModuleHomomorphism</code> is a synonym for the meet of <code class="func">IsLeftModuleGeneralMapping</code> and <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>).</p>

<p><a id="X7F6841107E59107F" name="X7F6841107E59107F"></a></p>

<h5>32.11-3 IsLinearMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsLinearMapping</code>( <var class="Arg">F</var>, <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>For a field <var class="Arg">F</var> and a general mapping <var class="Arg">mapp</var>, <code class="func">IsLinearMapping</code> returns <code class="keyw">true</code> if <var class="Arg">mapp</var> is an <var class="Arg">F</var>-linear mapping, and <code class="keyw">false</code> otherwise.</p>

<p>A mapping <span class="SimpleMath">f</span> is a linear mapping (or vector space homomorphism) if the source and range are vector spaces over the same division ring <span class="SimpleMath">D</span>, and if <span class="SimpleMath">f( a + b ) = f(a) + f(b)</span> and <span class="SimpleMath">f( s * a ) = s * f(a)</span> hold for all elements <span class="SimpleMath">a</span>, <span class="SimpleMath">b</span> in the source of <span class="SimpleMath">f</span> and <span class="SimpleMath">s ∈ D</span>.</p>

<p><a id="X7E88C32A82E942DA" name="X7E88C32A82E942DA"></a></p>

<h4>32.12 <span class="Heading">Ring Homomorphisms</span></h4>

<p><a id="X7C8DA031799B79D5" name="X7C8DA031799B79D5"></a></p>

<h5>32.12-1 IsRingGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRingGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRingHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p><code class="func">IsRingGeneralMapping</code> specifies whether a general mapping <var class="Arg">mapp</var> satisfies the property <code class="func">IsAdditiveGroupGeneralMapping</code> (<a href="chap32.html#X7B99EF287A8A0BD9"><span class="RefLink">32.10-4</span></a>) and respects multiplication (see <code class="func">RespectsMultiplication</code> (<a href="chap32.html#X7BEFF95883EAEC78"><span class="RefLink">32.9-1</span></a>)).</p>

<p><code class="func">IsRingHomomorphism</code> is a synonym for the meet of <code class="func">IsRingGeneralMapping</code> and <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>).</p>

<p><a id="X7988102883675606" name="X7988102883675606"></a></p>

<h5>32.12-2 IsRingWithOneGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRingWithOneGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRingWithOneHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p><a id="X86B14F908601DEA9" name="X86B14F908601DEA9"></a></p>

<h5>32.12-3 IsAlgebraGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsAlgebraGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsAlgebraHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p><code class="func">IsAlgebraGeneralMapping</code> specifies whether a general mapping <var class="Arg">mapp</var> satisfies both properties <code class="func">IsRingGeneralMapping</code> (<a href="chap32.html#X7C8DA031799B79D5"><span class="RefLink">32.12-1</span></a>) and (see <code class="func">IsLeftModuleGeneralMapping</code> (<a href="chap32.html#X780BE6307A3271A9"><span class="RefLink">32.11-2</span></a>)).</p>

<p><code class="func">IsAlgebraHomomorphism</code> is a synonym for the meet of <code class="func">IsAlgebraGeneralMapping</code> and <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>).</p>

<p><a id="X842AD44679C5BDC2" name="X842AD44679C5BDC2"></a></p>

<h5>32.12-4 IsAlgebraWithOneGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsAlgebraWithOneGeneralMapping</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsAlgebraWithOneHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;filter&nbsp;)</td></tr></table></div>
<p><code class="func">IsAlgebraWithOneGeneralMapping</code> specifies whether a general mapping <var class="Arg">mapp</var> satisfies both properties <code class="func">IsAlgebraGeneralMapping</code> (<a href="chap32.html#X86B14F908601DEA9"><span class="RefLink">32.12-3</span></a>) and <code class="func">RespectsOne</code> (<a href="chap32.html#X7EE4DA097AE9CBC1"><span class="RefLink">32.9-2</span></a>).</p>

<p><code class="func">IsAlgebraWithOneHomomorphism</code> is a synonym for the meet of <code class="func">IsAlgebraWithOneGeneralMapping</code> and <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>).</p>

<p><a id="X8324DA78879DF4D7" name="X8324DA78879DF4D7"></a></p>

<h5>32.12-5 IsFieldHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsFieldHomomorphism</code>( <var class="Arg">mapp</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>A general mapping is a field homomorphism if and only if it is a ring homomorphism with source a field.</p>

<p><a id="X7E4A55567BED0F88" name="X7E4A55567BED0F88"></a></p>

<h4>32.13 <span class="Heading">General Mappings</span></h4>

<p><a id="X8656AB8A7D672CAE" name="X8656AB8A7D672CAE"></a></p>

<h5>32.13-1 IsGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;category&nbsp;)</td></tr></table></div>
<p>Each general mapping lies in the category <code class="func">IsGeneralMapping</code>. It implies the categories <code class="func">IsMultiplicativeElementWithInverse</code> (<a href="chap31.html#X7FDB14E57814FA3B"><span class="RefLink">31.14-13</span></a>) and <code class="func">IsAssociativeElement</code> (<a href="chap31.html#X7979AFAA80FF795A"><span class="RefLink">31.15-1</span></a>); for a discussion of these implications, see <a href="chap32.html#X7E2E16277940FA0B"><span class="RefLink">32.6</span></a>.</p>

<p><a id="X791690817E23D90C" name="X791690817E23D90C"></a></p>

<h5>32.13-2 IsConstantTimeAccessGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsConstantTimeAccessGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>is <code class="keyw">true</code> if the underlying relation of the general mapping <var class="Arg">map</var> knows its <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) value, and <code class="keyw">false</code> otherwise.</p>

<p>In the former case, <var class="Arg">map</var> is allowed to use this list for calls to <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) etc.</p>

<p><a id="X81CFF5F87BBEA8AD" name="X81CFF5F87BBEA8AD"></a></p>

<h5>32.13-3 IsEndoGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsEndoGeneralMapping</code>( <var class="Arg">obj</var> )</td><td class="tdright">(&nbsp;property&nbsp;)</td></tr></table></div>
<p>If a general mapping has this property then its source and range are equal.</p>

<p><a id="X7D6F78587C00CDD0" name="X7D6F78587C00CDD0"></a></p>

<h4>32.14 <span class="Heading">Technical Matters Concerning General Mappings</span></h4>

<p><code class="func">Source</code> (<a href="chap32.html#X7DE8173F80E07AB1"><span class="RefLink">32.3-8</span></a>) and <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>) are basic operations for general mappings. <code class="func">UnderlyingRelation</code> (<a href="chap32.html#X784F871383FB599B"><span class="RefLink">32.3-9</span></a>) 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>

<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 <code class="func">IsNonSPGeneralMapping</code> (<a href="chap32.html#X7D28581F82481163"><span class="RefLink">32.14-1</span></a>), <code class="func">IsSPGeneralMapping</code> (<a href="chap32.html#X7D28581F82481163"><span class="RefLink">32.14-1</span></a>). (The category <code class="func">IsGeneralMapping</code> (<a href="chap32.html#X8656AB8A7D672CAE"><span class="RefLink">32.13-1</span></a>) is defined as the disjoint union of these two.)</p>

<p>For general mappings of the first category, <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) and <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>) are basic operations. (Note that in principle it is possible to delegate from <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>) to <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>).) Methods for the secondary operations <code class="func">ImageElm</code> (<a href="chap32.html#X7CFAB0157BFB1806"><span class="RefLink">32.4-5</span></a>), <code class="func">PreImageElm</code> (<a href="chap32.html#X7D212F727CAE971A"><span class="RefLink">32.5-3</span></a>), <code class="func">ImagesSet</code> (<a href="chap32.html#X8781348F7F5796A0"><span class="RefLink">32.4-4</span></a>), <code class="func">PreImagesSet</code> (<a href="chap32.html#X856BAFC87B2D2811"><span class="RefLink">32.5-5</span></a>), <code class="func">ImagesRepresentative</code> (<a href="chap32.html#X85ADB89B7C8DD7D0"><span class="RefLink">32.4-2</span></a>), and <code class="func">PreImagesRepresentative</code> (<a href="chap32.html#X7AE24A1586B7DE79"><span class="RefLink">32.5-4</span></a>) may use <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) and <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>), respectively, and methods for <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>), <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>) must <em>not</em> call the secondary operations. In particular, there are no generic methods for <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) and <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>).</p>

<p>Methods for <code class="func">ImagesSet</code> (<a href="chap32.html#X8781348F7F5796A0"><span class="RefLink">32.4-4</span></a>) and <code class="func">PreImagesSet</code> (<a href="chap32.html#X856BAFC87B2D2811"><span class="RefLink">32.5-5</span></a>) must <em>not</em> use <code class="func">PreImagesRange</code> (<a href="chap32.html#X78EF1FE77B0973C0"><span class="RefLink">32.5-1</span></a>) and <code class="func">ImagesSource</code> (<a href="chap32.html#X7D23C1CE863DACD8"><span class="RefLink">32.4-1</span></a>), e.g., compute the intersection of the set in question with the preimage of the range resp. the image of the source.</p>

<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 <code class="func">ImagesRepresentative</code> (<a href="chap32.html#X85ADB89B7C8DD7D0"><span class="RefLink">32.4-2</span></a>), <code class="func">PreImagesRepresentative</code> (<a href="chap32.html#X7AE24A1586B7DE79"><span class="RefLink">32.5-4</span></a>), <code class="func">KernelOfMultiplicativeGeneralMapping</code> (<a href="chap32.html#X81A5A5CF846E5FBF"><span class="RefLink">32.9-5</span></a>), and <code class="func">CoKernelOfMultiplicativeGeneralMapping</code> (<a href="chap32.html#X7F09B6E28080DCB4"><span class="RefLink">32.9-6</span></a>) as basic operations here, and to make <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) and <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>) secondary operations that may delegate to these.</p>

<p>In order to avoid infinite recursions, we must distinguish between the two different types of mappings.</p>

<p>(Note that the basic domain operations such as <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) for the underlying relation of a general mapping may use either <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) or <code class="func">ImagesRepresentative</code> (<a href="chap32.html#X85ADB89B7C8DD7D0"><span class="RefLink">32.4-2</span></a>) and the appropriate cokernel. Conversely, if <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) for the underlying relation is known then <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>) resp. <code class="func">ImagesRepresentative</code> (<a href="chap32.html#X85ADB89B7C8DD7D0"><span class="RefLink">32.4-2</span></a>) may delegate to it, the general mapping gets the property <code class="func">IsConstantTimeAccessGeneralMapping</code> (<a href="chap32.html#X791690817E23D90C"><span class="RefLink">32.13-2</span></a>) for this; note that this is not allowed if only an enumerator of the underlying relation is known.)</p>

<p>Secondary operations are <code class="func">IsInjective</code> (<a href="chap32.html#X7F065FD7822C0A12"><span class="RefLink">32.3-4</span></a>), <code class="func">IsSingleValued</code> (<a href="chap32.html#X86D44C8A78BF1981"><span class="RefLink">32.3-2</span></a>), <code class="func">IsSurjective</code> (<a href="chap32.html#X784ECE847E005B8F"><span class="RefLink">32.3-5</span></a>), <code class="func">IsTotal</code> (<a href="chap32.html#X83C7494E828CC9C8"><span class="RefLink">32.3-1</span></a>); they may use the basic operations, and must not be used by them.</p>

<p>Methods for the operations <code class="func">ImagesElm</code> (<a href="chap32.html#X7D51184B7EE5B2CF"><span class="RefLink">32.4-3</span></a>), <code class="func">ImagesRepresentative</code> (<a href="chap32.html#X85ADB89B7C8DD7D0"><span class="RefLink">32.4-2</span></a>), <code class="func">ImagesSet</code> (<a href="chap32.html#X8781348F7F5796A0"><span class="RefLink">32.4-4</span></a>), <code class="func">ImageElm</code> (<a href="chap32.html#X7CFAB0157BFB1806"><span class="RefLink">32.4-5</span></a>), <code class="func">PreImagesElm</code> (<a href="chap32.html#X7FBB830C8729E995"><span class="RefLink">32.5-2</span></a>), <code class="func">PreImagesRepresentative</code> (<a href="chap32.html#X7AE24A1586B7DE79"><span class="RefLink">32.5-4</span></a>), <code class="func">PreImagesSet</code> (<a href="chap32.html#X856BAFC87B2D2811"><span class="RefLink">32.5-5</span></a>), and <code class="func">PreImageElm</code> (<a href="chap32.html#X7D212F727CAE971A"><span class="RefLink">32.5-3</span></a>) take two arguments, a general mapping <var class="Arg">map</var> and an element or collection of elements <var class="Arg">elm</var>. These methods must <em>not</em> check whether <var class="Arg">elm</var> lies in the source or the range of <var class="Arg">map</var>. In the case that <var class="Arg">elm</var> does not, <code class="keyw">fail</code> may be returned as well as any other <strong class="pkg">GAP</strong> object, and even an error message is allowed. Checks of the arguments are done only by the functions <code class="func">Image</code> (<a href="chap32.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>), <code class="func">Images</code> (<a href="chap32.html#X86114B2E7E77488C"><span class="RefLink">32.4-7</span></a>), <code class="func">PreImage</code> (<a href="chap32.html#X836FAEAC78B55BF4"><span class="RefLink">32.5-6</span></a>), and <code class="func">PreImages</code> (<a href="chap32.html#X85C8590E832002EF"><span class="RefLink">32.5-7</span></a>), which then delegate to the operations listed above.</p>

<p><a id="X7D28581F82481163" name="X7D28581F82481163"></a></p>

<h5>32.14-1 IsSPGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSPGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;category&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsNonSPGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">(&nbsp;category&nbsp;)</td></tr></table></div>
<p><a id="X80D02AD183E01F16" name="X80D02AD183E01F16"></a></p>

<h5>32.14-2 IsGeneralMappingFamily</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsGeneralMappingFamily</code>( <var class="Arg">obj</var> )</td><td class="tdright">(&nbsp;category&nbsp;)</td></tr></table></div>
<p>The family category of the category of general mappings.</p>

<p><a id="X86CFADBA7F2FE446" name="X86CFADBA7F2FE446"></a></p>

<h5>32.14-3 FamilyRange</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FamilyRange</code>( <var class="Arg">Fam</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>is the elements family of the family of the range of each general mapping in the family <var class="Arg">Fam</var>.</p>

<p><a id="X7C3736E281A9E505" name="X7C3736E281A9E505"></a></p>

<h5>32.14-4 FamilySource</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FamilySource</code>( <var class="Arg">Fam</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>is the elements family of the family of the source of each general mapping in the family <var class="Arg">Fam</var>.</p>

<p><a id="X7AE54FB67E2E6374" name="X7AE54FB67E2E6374"></a></p>

<h5>32.14-5 FamiliesOfGeneralMappingsAndRanges</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FamiliesOfGeneralMappingsAndRanges</code>( <var class="Arg">Fam</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>is a list that stores at the odd positions the families of general mappings with source in the family <var class="Arg">Fam</var>, at the even positions the families of ranges of the general mappings.</p>

<p><a id="X7E1E26E37C413F6F" name="X7E1E26E37C413F6F"></a></p>

<h5>32.14-6 GeneralMappingsFamily</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneralMappingsFamily</code>( <var class="Arg">sourcefam</var>, <var class="Arg">rangefam</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>All general mappings with same source family <var class="Arg">FS</var> and same range family <var class="Arg">FR</var> lie in the family <code class="code">GeneralMappingsFamily( <var class="Arg">FS</var>, <var class="Arg">FR</var> )</code>.</p>

<p><a id="X7CF92CC37A6BBDA5" name="X7CF92CC37A6BBDA5"></a></p>

<h5>32.14-7 TypeOfDefaultGeneralMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; TypeOfDefaultGeneralMapping</code>( <var class="Arg">source</var>, <var class="Arg">range</var>, <var class="Arg">filter</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>is the type of mappings with <code class="code">IsDefaultGeneralMappingRep</code> with source <var class="Arg">source</var> and range <var class="Arg">range</var> and additional categories <var class="Arg">filter</var>.</p>


<div class="chlinkprevnextbot">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap31.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap33.html">[Next Chapter]</a>&nbsp;  </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>