1
  
2
  3 Basic operations with numerical semigroups
3
  
4
  
5
  3.1 Invariants
6
  
7
  3.1-1 Multiplicity
8
  
9
  Multiplicity( NS )  attribute
10
  MultiplicityOfNumericalSemigroup( NS )  attribute
11
  
12
  NS  is  a  numerical semigroup. Returns the multiplicity of NS, which is the
13
  smallest positive integer belonging to NS.
14
  
15
    Example  
16
    gap> S := NumericalSemigroup("modular", 7,53);
17
    <Modular numerical semigroup satisfying 7x mod 53 <= x >
18
    gap> MultiplicityOfNumericalSemigroup(S);
19
    8
20
    gap> NumericalSemigroup(3,5);
21
    <Numerical semigroup with 2 generators>
22
    gap> Multiplicity(last);
23
    3
24
  
25
  
26
  3.1-2 GeneratorsOfNumericalSemigroup
27
  
28
  GeneratorsOfNumericalSemigroup( S )  attribute
29
  Generators( S )  attribute
30
  MinimalGeneratingSystemOfNumericalSemigroup( S )  attribute
31
  MinimalGeneratingSystem( S )  attribute
32
  MinimalGenerators( S )  attribute
33
  
34
  S  is a numerical semigroup. GeneratorsOfNumericalSemigroup returns a set of
35
  generators      of      S,      which      may      not      be     minimal.
36
  MinimalGeneratingSystemOfNumericalSemigroup   returns  the  minimal  set  of
37
  generators of S.
38
  
39
  From  Version 0.980, ReducedSetOfGeneratorsOfNumericalSemigroup is a synonym
40
  of                              MinimalGeneratingSystemOfNumericalSemigroup;
41
  GeneratorsOfNumericalSemigroupNC        is        a        synonym        of
42
  GeneratorsOfNumericalSemigroup.  The  names  are kept for compatibility with
43
  code produced for previous versions, but will be removed in the future.
44
  
45
    Example  
46
    gap> S := NumericalSemigroup("modular", 5,53);
47
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
48
    gap> GeneratorsOfNumericalSemigroup(S);
49
    [ 11, 12, 13, 32, 53 ]
50
    gap> S := NumericalSemigroup(3, 5, 53);
51
    <Numerical semigroup with 3 generators>
52
    gap> GeneratorsOfNumericalSemigroup(S);
53
    [ 3, 5, 53 ]
54
    gap> MinimalGeneratingSystemOfNumericalSemigroup(S);
55
    [ 3, 5 ]
56
    gap> MinimalGeneratingSystem(S)=MinimalGeneratingSystemOfNumericalSemigroup(S);
57
    true
58
    gap> s := NumericalSemigroup(3,5,7,15);
59
    <Numerical semigroup with 4 generators>
60
    gap> HasGenerators(s);
61
    true
62
    gap> HasMinimalGenerators(s);
63
    false
64
    gap> MinimalGenerators(s);
65
    [ 3, 5, 7 ]
66
    gap> Generators(s);
67
    [ 3, 5, 7, 15 ]
68
  
69
  
70
  3.1-3 EmbeddingDimension
71
  
72
  EmbeddingDimension( NS )  attribute
73
  EmbeddingDimensionOfNumericalSemigroup( NS )  attribute
74
  
75
  NS  is  a  numerical  semigroup.  It  returns the cardinality of its minimal
76
  generating system.
77
  
78
    Example  
79
    gap> s := NumericalSemigroup(3,5,7,15);
80
    <Numerical semigroup with 4 generators>
81
    gap> EmbeddingDimension(s);
82
    3
83
    gap> EmbeddingDimensionOfNumericalSemigroup(s);
84
    3
85
  
86
  
87
  3.1-4 SmallElements
88
  
89
  SmallElements( NS )  attribute
90
  SmallElementsOfNumericalSemigroup( NS )  attribute
91
  
92
  NS is a numerical semigroup. It returns the list of small elements of NS. Of
93
  course,  the  time  consumed  to  return  a result may depend on the way the
94
  semigroup is given.
95
  
96
    Example  
97
    gap> SmallElementsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
98
    [ 0, 3, 5 ]
99
    gap> SmallElements(NumericalSemigroup(3,5,7));
100
    [ 0, 3, 5 ]
101
  
102
  
103
  3.1-5 FirstElementsOfNumericalSemigroup
104
  
105
  FirstElementsOfNumericalSemigroup( n, NS )  function
106
  
107
  NS  is  a numerical semigroup. It returns the list with the first n elements
108
  of NS.
109
  
110
    Example  
111
    gap> FirstElementsOfNumericalSemigroup(2,NumericalSemigroup(3,5,7));
112
    [ 0, 3 ]
113
    gap> FirstElementsOfNumericalSemigroup(10,NumericalSemigroup(3,5,7));
114
    [ 0, 3, 5, 6, 7, 8, 9, 10, 11, 12 ]
115
  
116
  
117
  3.1-6 RthElementOfNumericalSemigroup
118
  
119
  RthElementOfNumericalSemigroup( S, r )  operation
120
  
121
  S  is a numerical semigroup and r is an integer. It returns the r-th element
122
  of S.
123
  
124
    Example  
125
    gap> S := NumericalSemigroup(7,8,17);;
126
    gap> RthElementOfNumericalSemigroup(S,53);
127
    68
128
  
129
  
130
  3.1-7 AperyList
131
  
132
  AperyList( S, n )  attribute
133
  AperyListOfNumericalSemigroupWRTElement( S, n )  operation
134
  
135
  S  is  a  numerical semigroup and n is a positive element of S. Computes the
136
  Apéry  list of S with respect to n. It contains for every i∈ {0,...,n-1}, in
137
  the  i+1th  position, the smallest element in the semigroup congruent with i
138
  modulo n.
139
  
140
    Example  
141
    gap> S := NumericalSemigroup("modular", 5,53);;
142
    gap> AperyListOfNumericalSemigroupWRTElement(S,12);
143
    [ 0, 13, 26, 39, 52, 53, 54, 43, 32, 33, 22, 11 ]
144
    gap> AperyList(S,12);
145
    [ 0, 13, 26, 39, 52, 53, 54, 43, 32, 33, 22, 11 ]
146
  
147
  
148
  3.1-8 AperyList
149
  
150
  AperyList( S )  attribute
151
  AperyListOfNumericalSemigroup( S )  attribute
152
  
153
  S  is a numerical semigroup. It computes the Apéry list of S with respect to
154
  the multiplicity of S.
155
  
156
    Example  
157
    gap> S := NumericalSemigroup("modular", 5,53);;
158
    gap> AperyListOfNumericalSemigroup(S);
159
    [ 0, 12, 13, 25, 26, 38, 39, 51, 52, 53, 32 ]
160
    gap> AperyList(NumericalSemigroup(5,7,11));
161
    [ 0, 11, 7, 18, 14 ]
162
  
163
  
164
  3.1-9 AperyList
165
  
166
  AperyList( S, n )  attribute
167
  AperyListOfNumericalSemigroupWRTInteger( S, m )  function
168
  
169
  S is a numerical semigroup and m is an integer. Computes the Apéry list of S
170
  with  respect to m, that is, the set of elements x in S such that x-m is not
171
  in    S.    If    m   is   an   element   in   S,   then   the   output   of
172
  AperyListOfNumericalSemigroupWRTInteger,   as   sets,   is   the   same   as
173
  AperyListOfNumericalSemigroupWRTElement, though without side effects, in the
174
  sense  that this information is no longer used by the package. The output of
175
  AperyList is the same as AperyListOfNumericalSemigroupWRTElement.
176
  
177
    Example  
178
    gap>  s:=NumericalSemigroup(10,13,19,27);;
179
    gap> AperyListOfNumericalSemigroupWRTInteger(s,11);
180
    [ 0, 10, 13, 19, 20, 23, 26, 27, 29, 32, 33, 36, 39, 42, 45, 46, 52, 55 ]
181
    gap> AperyList(s,11);
182
    [ 0, 10, 13, 19, 20, 23, 26, 27, 29, 32, 33, 36, 39, 42, 45, 46, 52, 55 ]
183
    gap> Length(last);
184
    18
185
    gap> AperyListOfNumericalSemigroupWRTInteger(s,10);
186
    [ 0, 13, 19, 26, 27, 32, 38, 45, 51, 54 ]
187
    gap> AperyListOfNumericalSemigroupWRTElement(s,10);
188
    [ 0, 51, 32, 13, 54, 45, 26, 27, 38, 19 ]
189
    gap> Length(last);
190
    10
191
    gap> AperyList(s,10);
192
    [ 0, 51, 32, 13, 54, 45, 26, 27, 38, 19 ]
193
  
194
  
195
  3.1-10 AperyListOfNumericalSemigroupAsGraph
196
  
197
  AperyListOfNumericalSemigroupAsGraph( ap )  function
198
  
199
  ap  is  the  Apéry  list of a numerical semigroup. This function returns the
200
  adjacency  list of the graph (ap, E) where the edge u -> v is in E iff v - u
201
  is in ap. The 0 is ignored.
202
  
203
    Example  
204
    gap> s:=NumericalSemigroup(3,7);;
205
    gap> AperyListOfNumericalSemigroupWRTElement(s,10);
206
    [ 0, 21, 12, 3, 14, 15, 6, 7, 18, 9 ]
207
    gap> AperyListOfNumericalSemigroupAsGraph(last);
208
    [ ,, [ 3, 6, 9, 12, 15, 18, 21 ],,, [ 6, 9, 12, 15, 18, 21 ],
209
    [ 7, 14, 21 ],, [ 9, 12, 15, 18, 21 ],,, [ 12, 15, 18, 21 ],,
210
    [ 14, 21 ], [ 15, 18, 21 ],,, [ 18, 21 ],,, [ 21 ] ]
211
  
212
  
213
  3.1-11 KunzCoordinatesOfNumericalSemigroup
214
  
215
  KunzCoordinatesOfNumericalSemigroup( S, m )  function
216
  
217
  S  is  a  numerical  semigroup,  and m is a nonzero element of S. The second
218
  argument is optional, and if missing it is assumed to be the multiplicity of
219
  S.
220
  
221
  Then the Apéry set of m in S has the form [0,k_1m+1,...,k_m-1m+m-1], and the
222
  output is the (m-1)-uple [k_1,k_2,...,k_m-1]
223
  
224
    Example  
225
    gap> s:=NumericalSemigroup(3,5,7);
226
    <Numerical semigroup with 3 generators>
227
    gap> KunzCoordinatesOfNumericalSemigroup(s);
228
    [ 2, 1 ]
229
    gap> KunzCoordinatesOfNumericalSemigroup(s,5);
230
    [ 1, 1, 0, 1 ]
231
  
232
  
233
  3.1-12 KunzPolytope
234
  
235
  KunzPolytope( m )  function
236
  
237
  m is a positive integer.
238
  
239
  The  Kunz coordinates of the semigroups with multiplicity m are solutions of
240
  a  system  of  inequalities Axge b (see [CAGGB02]). The output is the matrix
241
  (A|-b).
242
  
243
    Example  
244
    gap> KunzPolytope(3);
245
    [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 2, -1, 0 ], [ -1, 2, 1 ] ]
246
  
247
  
248
  3.1-13 CocycleOfNumericalSemigroupWRTElement
249
  
250
  CocycleOfNumericalSemigroupWRTElement( S, m )  function
251
  
252
  S  is  a numerical semigroup, and m is a nonzero element of S. The output is
253
  the  matrix  h(i,j)=(w(i)+w(j)-w((i+j)mod  m))/m, where w(i) is the smallest
254
  element  in  S congruent with i modulo m (and thus it is in the Apéry set of
255
  m), [GSHKR17].
256
  
257
    Example  
258
    gap> s:=NumericalSemigroup(3,5,7);;
259
    gap> CocycleOfNumericalSemigroupWRTElement(s,3);
260
    [ [ 0, 0, 0 ], [ 0, 3, 4 ], [ 0, 4, 1 ] ]
261
  
262
  
263
  3.1-14 FrobeniusNumber
264
  
265
  FrobeniusNumber( NS )  attribute
266
  FrobeniusNumberOfNumericalSemigroup( NS )  attribute
267
  
268
  The  largest nonnegative integer not belonging to a numerical semigroup S is
269
  the  Frobenius  number  of  S. If S is the set of nonnegative integers, then
270
  clearly its Frobenius number is -1, otherwise its Frobenius number coincides
271
  with the maximum of the gaps (or fundamental gaps) of S.
272
  
273
  NS  is  a  numerical  semigroup.  It  returns the Frobenius number of NS. Of
274
  course,  the  time  consumed  to  return  a result may depend on the way the
275
  semigroup is given or on the knowledge already produced on the semigroup.
276
  
277
    Example  
278
    gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7));
279
    4
280
    gap> FrobeniusNumber(NumericalSemigroup(3,5,7));
281
    4
282
  
283
  
284
  3.1-15 Conductor
285
  
286
  Conductor( NS )  attribute
287
  ConductorOfNumericalSemigroup( NS )  attribute
288
  
289
  This is just a synonym of  FrobeniusNumberOfNumericalSemigroup (NS)+1.
290
  
291
    Example  
292
    gap> ConductorOfNumericalSemigroup(NumericalSemigroup(3,5,7));
293
    5
294
    gap> Conductor(NumericalSemigroup(3,5,7));
295
    5
296
  
297
  
298
  3.1-16 PseudoFrobeniusOfNumericalSemigroup
299
  
300
  PseudoFrobeniusOfNumericalSemigroup( S )  attribute
301
  
302
  An integer z is a pseudo-Frobenius number of S if z+S∖{0}⊆ S.
303
  
304
  S is a numerical semigroup. It returns set of pseudo-Frobenius numbers of S.
305
  
306
    Example  
307
    gap> S := NumericalSemigroup("modular", 5,53);
308
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
309
    gap> PseudoFrobeniusOfNumericalSemigroup(S);
310
    [ 21, 40, 41, 42 ]
311
  
312
  
313
  3.1-17 TypeOfNumericalSemigroup
314
  
315
  TypeOfNumericalSemigroup( NS )  attribute
316
  
317
  Stands for Length(PseudoFrobeniusOfNumericalSemigroup (NS)).
318
  
319
    Example  
320
    gap> S := NumericalSemigroup("modular", 5,53);
321
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
322
    gap> Type(S);
323
    4
324
    gap> TypeOfNumericalSemigroup(S);
325
    4
326
  
327
  
328
  3.1-18 Gaps
329
  
330
  Gaps( NS )  attribute
331
  GapsOfNumericalSemigroup( NS )  attribute
332
  
333
  A  gap  of a numerical semigroup S is a nonnegative integer not belonging to
334
  S. NS is a numerical semigroup. Both return the set of gaps of NS.
335
  
336
    Example  
337
    gap> GapsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
338
    [ 1, 2, 4 ]
339
    gap> Gaps(NumericalSemigroup(5,7,11));
340
    [ 1, 2, 3, 4, 6, 8, 9, 13 ]
341
  
342
  
343
  3.1-19 DesertsOfNumericalSemigroup
344
  
345
  DesertsOfNumericalSemigroup( NS )  function
346
  
347
  NS is a numerical semigroup. The output is the list with the runs of gaps of
348
  NS.
349
  
350
    Example  
351
    gap> s:=NumericalSemigroup(3,5,7);;
352
    gap> DesertsOfNumericalSemigroup(s);
353
    [ [ 1, 2 ], [ 4 ] ]
354
  
355
  
356
  3.1-20 IsOrdinaryNumericalSemigroup
357
  
358
  IsOrdinaryNumericalSemigroup( NS )  property
359
  IsOrdinary( NS )  property
360
  
361
  NS is a numerical semigroup. Dectects if the semigroup is ordinary, that is,
362
  with less than two deserts.
363
  
364
    Example  
365
    gap> s:=NumericalSemigroup(3,5,7);;
366
    gap> IsOrdinary(s);
367
    false
368
  
369
  
370
  3.1-21 IsAcuteNumericalSemigroup
371
  
372
  IsAcuteNumericalSemigroup( NS )  property
373
  IsAcute( NS )  property
374
  
375
  NS is a numerical semigroup. Dectects if the semigroup is acute, that is, it
376
  is  either  ordinary  or its last desert (the one with the Frobenius number)
377
  has less elements than the preceding one ([BA04]).
378
  
379
    Example  
380
    gap> s:=NumericalSemigroup(3,5,7);;
381
    gap> IsAcute(s);
382
    true
383
  
384
  
385
  3.1-22 Holes
386
  
387
  Holes( NS )  attribute
388
  HolesOfNumericalSemigroup( S )  attribute
389
  
390
  S  is a numerical semigroup. Returns the set of gaps x of S such that F(S)-x
391
  is also a gap, where F(S) stands for the Frobenius number of S.
392
  
393
    Example  
394
    gap> s:=NumericalSemigroup(3,5);;
395
    gap> Holes(s);
396
    [  ]
397
    gap> s:=NumericalSemigroup(3,5,7);;
398
    gap> HolesOfNumericalSemigroup(s);
399
    [ 2 ]
400
  
401
  
402
  3.1-23 LatticePathAssociatedToNumericalSemigroup
403
  
404
  LatticePathAssociatedToNumericalSemigroup( S, p, q )  attribute
405
  
406
  S is a numerical semigroup and p,q are two elements in S.
407
  
408
  In  this setting S is an oversemigroup of ⟨ p,q⟩, and consequently every gap
409
  of  S  is a gap of ⟨ p,q⟩. If c is the conductor of ⟨ p,q⟩, then every gap g
410
  of  ⟨  p,q⟩  can  be  written uniquely as g=c-1-(ap+bp) for some nonnegative
411
  integers a,b. We say that (a,b) are the coordinates associated to g.
412
  
413
  The  output  is  a  path  in  N^2  such  that  coordinates  of the gaps of S
414
  correspond  exactly  with the points in N^2 that are between the path in the
415
  line ax+by=c-1. See [KW14].
416
  
417
    Example  
418
    gap> s:=NumericalSemigroup(16,17,71,72);;
419
    gap> LatticePathAssociatedToNumericalSemigroup(s,16,17);
420
    [ [ 0, 14 ], [ 1, 13 ], [ 2, 12 ], [ 3, 11 ], [ 4, 10 ], [ 5, 9 ], [ 6, 8 ],
421
      [ 7, 7 ], [ 8, 6 ], [ 9, 5 ], [ 10, 4 ], [ 11, 3 ], [ 12, 2 ], [ 13, 1 ],
422
      [ 14, 0 ] ]
423
  
424
  
425
  3.1-24 Genus
426
  
427
  Genus( NS )  attribute
428
  GenusOfNumericalSemigroup( NS )  attribute
429
  
430
  NS is a numerical semigroup. It returns the number of gaps of NS.
431
  
432
    Example  
433
    gap> s:=NumericalSemigroup(16,17,71,72);;
434
    gap> GenusOfNumericalSemigroup(s);
435
    80
436
    gap> S := NumericalSemigroup("modular", 5,53);
437
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
438
    gap> Genus(S);
439
    26
440
  
441
  
442
  3.1-25 FundamentalGaps
443
  
444
  FundamentalGaps( S )  attribute
445
  FundamentalGapsOfNumericalSemigroup( S )  attribute
446
  
447
  S  The fundamental gaps of S are those gaps that are maximal with respect to
448
  the  partial  order  induced  by division in N. is a numerical semigroup. It
449
  returns the set of fundamental gaps of S.
450
  
451
    Example  
452
    gap> S := NumericalSemigroup("modular", 5,53);
453
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
454
    gap> FundamentalGapsOfNumericalSemigroup(S);
455
    [ 16, 17, 18, 19, 27, 28, 29, 30, 31, 40, 41, 42 ]
456
    gap> GapsOfNumericalSemigroup(S);
457
    [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29,
458
      30, 31, 40, 41, 42 ]
459
    gap> Gaps(NumericalSemigroup(5,7,11));
460
    [ 1, 2, 3, 4, 6, 8, 9, 13 ]
461
    gap> FundamentalGaps(NumericalSemigroup(5,7,11));
462
    [ 6, 8, 9, 13 ]
463
  
464
  
465
  3.1-26 SpecialGaps
466
  
467
  SpecialGaps( S )  attribute
468
  SpecialGapsOfNumericalSemigroup( S )  attribute
469
  
470
  The special gaps of a numerical semigroup S, are those fundamental gaps such
471
  that  if they are added to the given numerical semigroup, then the resulting
472
  set  is  again a numerical semigroup. S is a numerical semigroup. It returns
473
  the special gaps of S.
474
  
475
    Example  
476
    gap> S := NumericalSemigroup("modular", 5,53);
477
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
478
    gap> SpecialGaps(S);
479
    [ 40, 41, 42 ]
480
    gap> SpecialGapsOfNumericalSemigroup(S);
481
    [ 40, 41, 42 ]
482
  
483
  
484
  
485
  3.2 Wilf's conjecture
486
  
487
  Let  S be a numerical semigroup, with conductor c and embedding dimension e.
488
  Denote by l the cardinality of the set of elements in S smaller than c. Wilf
489
  in  [Wil78]  asked whether or not l/cge 1/e for all numerical semigroups. In
490
  this  section  we give some functions to experiment with this conjecture, as
491
  defined in [Eli15].
492
  
493
  3.2-1 WilfNumber
494
  
495
  WilfNumber( S )  attribute
496
  WilfNumberOfNumericalSemigroup( S )  attribute
497
  
498
  S  is  a  numerical  semigroup.  Let  c, e and l be the conductor, embedding
499
  dimension  and  number of elements smaller than c in S. Returns e l-c, which
500
  was conjetured by Wilf to be nonnegative.
501
  
502
    Example  
503
    gap> l:=NumericalSemigroupsWithGenus(10);;
504
    gap> Filtered(l, s->WilfNumberOfNumericalSemigroup(s)<0); 
505
    [  ]
506
    gap> Maximum(Set(l, s->WilfNumberOfNumericalSemigroup(s)));
507
    70
508
    gap> s := NumericalSemigroup(13,25,37);;
509
    gap> WilfNumber(s);                     
510
    96
511
  
512
  
513
  3.2-2 EliahouNumber
514
  
515
  EliahouNumber( S )  attribute
516
  TruncatedWilfNumberOfNumericalSemigroup( S )  attribute
517
  
518
  S  is  a  numerical  semigroup.  Let  c,  m,  s  and  l  be  the  conductor,
519
  multiplicity,  number  of  generators smaller than c, and number of elements
520
  smaller  than c in S, respectively. Let q and r be the quotient and negative
521
  remainder  of  the  division of c by m, that is, c=qm-r. Returns s l-qd_q+r,
522
  where  d_q  corresponds  with the number of integers in [c,c+m[ that are not
523
  minimal generators of S.
524
  
525
    Example  
526
    gap> s:=NumericalSemigroup(5,7,9);;
527
    gap> TruncatedWilfNumberOfNumericalSemigroup(s);
528
    4
529
    gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;
530
    gap> EliahouNumber(s);
531
    -1
532
  
533
  
534
  3.2-3 ProfileOfNumericalSemigroup
535
  
536
  ProfileOfNumericalSemigroup( S )  attribute
537
  
538
  S is a numerical semigroup. Let c and m be the conductor and multiplicity of
539
  S,  respectively.  Let  q and r be the quotient and nonpositive remainder of
540
  the  division  of  c  by  m,  that  is,  c=qm-r.  Returns a list of lists of
541
  integers,  each  list  is  the cardinality of S ∩ [jm-r, (j+1)m-r[ with j in
542
  [1..q-1].
543
  
544
    Example  
545
    gap> s:=NumericalSemigroup(5,7,9);;
546
    gap> ProfileOfNumericalSemigroup(s);
547
    [ 2, 1 ]
548
    gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;
549
    gap> ProfileOfNumericalSemigroup(s);
550
    [ 3, 0, 0 ]
551
  
552
  
553
  3.2-4 EliahouSlicesOfNumericalSemigroup
554
  
555
  EliahouSlicesOfNumericalSemigroup( S )  attribute
556
  
557
  S is a numerical semigroup. Let c and m be the conductor and multiplicity of
558
  S,  respectively.  Let q and r be the quotient and negative remainder of the
559
  division  of  c  by m, that is, c=qm-r. Returns a list of lists of integers,
560
  each  list  is  the  set S ∩ [jm-r, (j+1)m-r[ with j in [1..q]. So this is a
561
  partition of the set of small elements of S (without 0).
562
  
563
    Example  
564
    gap> s:=NumericalSemigroup(5,7,9);;                                     
565
    gap> EliahouSlicesOfNumericalSemigroup(s);
566
    [ [ 5, 7 ], [ 9, 10, 12 ] ]
567
    gap> SmallElements(s);
568
    [ 0, 5, 7, 9, 10, 12, 14 ]
569
  
570
  
571

572