This repository has been archived by the owner on Jan 30, 2023. It is now read-only.
-
-
Notifications
You must be signed in to change notification settings - Fork 7
/
indexed_free_monoid.py
962 lines (785 loc) · 28.1 KB
/
indexed_free_monoid.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
"""
Indexed Monoids
AUTHORS:
- Travis Scrimshaw (2013-10-15)
"""
#*****************************************************************************
# Copyright (C) 2013 Travis Scrimshaw <tscrim at ucdavis.edu>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
from copy import copy
from sage.misc.abstract_method import abstract_method
from sage.misc.cachefunc import cached_method
from sage.structure.parent import Parent
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.element import MonoidElement
from sage.structure.indexed_generators import IndexedGenerators, parse_indices_names
from sage.structure.sage_object import op_EQ, op_NE, py_rich_to_bool
from sage.combinat.dict_addition import dict_addition
from sage.categories.monoids import Monoids
from sage.categories.poor_man_map import PoorManMap
from sage.categories.sets_cat import Sets
from sage.rings.integer import Integer
from sage.rings.infinity import infinity
from sage.rings.all import ZZ
from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
from sage.sets.family import Family
class IndexedMonoidElement(MonoidElement):
"""
An element of an indexed monoid.
This is an abstract class which uses the (abstract) method
:meth:`_sorted_items` for all of its functions. So to implement an
element of an indexed monoid, one just needs to implement
:meth:`_sorted_items`, which returns a list of pairs ``(i, p)`` where
``i`` is the index and ``p`` is the corresponding power, sorted in some
order. For example, in the free monoid there is no such choice, but for
the free abelian monoid, one could want lex order or have the highest
powers first.
Indexed monoid elements are ordered lexicographically with respect to
the result of :meth:`_sorted_items` (which for abelian free monoids is
influenced by the order on the indexing set).
"""
def __init__(self, F, x):
"""
Create the element ``x`` of an indexed free abelian monoid ``F``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F.gen(1)
F[1]
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: x = a^2 * b^3 * a^2 * b^4; x
F[0]^4*F[1]^7
sage: TestSuite(x).run()
sage: F = FreeMonoid(index_set=tuple('abcde'))
sage: a,b,c,d,e = F.gens()
sage: a in F
True
sage: a*b in F
True
sage: TestSuite(a*d^2*e*c*a).run()
"""
MonoidElement.__init__(self, F)
self._monomial = x
@abstract_method
def _sorted_items(self):
"""
Return the sorted items (i.e factors) of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: x = a*b^2*e*d
sage: x._sorted_items()
((0, 1), (1, 2), (4, 1), (3, 1))
.. SEEALSO::
:meth:`_repr_`, :meth:`_latex_`, :meth:`print_options`
"""
def _repr_(self):
"""
Return a string representation of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a*b^2*e*d
F[0]*F[1]^2*F[3]*F[4]
"""
if not self._monomial:
return '1'
monomial = self._sorted_items()
P = self.parent()
scalar_mult = P._print_options['scalar_mult']
exp = lambda v: '^{}'.format(v) if v != 1 else ''
return scalar_mult.join(P._repr_generator(g) + exp(v) for g,v in monomial)
def _ascii_art_(self):
r"""
Return an ASCII art representation of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: ascii_art(a*e*d)
F *F *F
0 3 4
sage: ascii_art(a*b^2*e*d)
2
F *F *F *F
0 1 3 4
"""
from sage.typeset.ascii_art import AsciiArt, ascii_art, empty_ascii_art
if not self._monomial:
return AsciiArt(["1"])
monomial = self._sorted_items()
P = self.parent()
scalar_mult = P._print_options['scalar_mult']
if all(x[1] == 1 for x in monomial):
ascii_art_gen = lambda m: P._ascii_art_generator(m[0])
else:
pref = AsciiArt([P.prefix()])
def ascii_art_gen(m):
if m[1] != 1:
r = (AsciiArt([" "**Integer(len(pref))]) + ascii_art(m[1]))
else:
r = empty_ascii_art
r = r * P._ascii_art_generator(m[0])
r._baseline = r._h - 2
return r
b = ascii_art_gen(monomial[0])
for x in monomial[1:]:
b = b + AsciiArt([scalar_mult]) + ascii_art_gen(x)
return b
def _latex_(self):
r"""
Return a `\LaTeX` representation of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: latex(a*b^2*e*d)
F_{0} F_{1}^{2} F_{3} F_{4}
"""
if not self._monomial:
return '1'
monomial = self._sorted_items()
P = self.parent()
scalar_mult = P._print_options['latex_scalar_mult']
if scalar_mult is None:
scalar_mult = P._print_options['scalar_mult']
if scalar_mult == "*":
scalar_mult = " "
exp = lambda v: '^{{{}}}'.format(v) if v != 1 else ''
return scalar_mult.join(P._latex_generator(g) + exp(v) for g,v in monomial)
def __iter__(self):
"""
Iterate over ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: list(b*a*c^3*b)
[(F[1], 1), (F[0], 1), (F[2], 3), (F[1], 1)]
::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: list(b*c^3*a)
[(F[0], 1), (F[1], 1), (F[2], 3)]
"""
return ((self.parent().gen(index), exp) for (index,exp) in self._sorted_items())
def _richcmp_(self, other, op):
r"""
Comparisons
TESTS::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a == a
True
sage: a*e == a*e
True
sage: a*b*c^3*b*d == (a*b*c)*(c^2*b*d)
True
sage: a != b
True
sage: a*b != b*a
True
sage: a*b*c^3*b*d != (a*b*c)*(c^2*b*d)
False
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a < b
True
sage: a*b < b*a
True
sage: a*b < a*a
False
sage: a^2*b < a*b*b
True
sage: b > a
True
sage: a*b > b*a
False
sage: a*b > a*a
True
sage: a*b <= b*a
True
sage: a*b <= b*a
True
sage: FA = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [FA.gen(i) for i in range(5)]
sage: a == a
True
sage: a*e == e*a
True
sage: a*b*c^3*b*d == a*d*(b^2*c^2)*c
True
sage: a != b
True
sage: a*b != a*a
True
sage: a*b*c^3*b*d != a*d*(b^2*c^2)*c
False
"""
if op == op_EQ:
return self._monomial == other._monomial
elif op == op_NE:
return self._monomial != other._monomial
return py_rich_to_bool(op, cmp(self.to_word_list(), other.to_word_list()))
def support(self):
"""
Return a list of the objects indexing ``self`` with
non-zero exponents.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (b*a*c^3*b).support()
[0, 1, 2]
::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (a*c^3).support()
[0, 2]
"""
supp = set([key for key, exp in self._sorted_items() if exp != 0])
return sorted(supp)
def leading_support(self):
"""
Return the support of the leading generator of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (b*a*c^3*a).leading_support()
1
::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (b*c^3*a).leading_support()
0
"""
if not self:
return None
return self._sorted_items()[0][0]
def trailing_support(self):
"""
Return the support of the trailing generator of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (b*a*c^3*a).trailing_support()
0
::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (b*c^3*a).trailing_support()
2
"""
if not self:
return None
return self._sorted_items()[-1][0]
def to_word_list(self):
"""
Return ``self`` as a word represented as a list whose entries
are indices of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (b*a*c^3*a).to_word_list()
[1, 0, 2, 2, 2, 0]
::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (b*c^3*a).to_word_list()
[0, 1, 2, 2, 2]
"""
return [k for k,e in self._sorted_items() for dummy in range(e)]
class IndexedFreeMonoidElement(IndexedMonoidElement):
"""
An element of an indexed free abelian monoid.
"""
def __init__(self, F, x):
"""
Create the element ``x`` of an indexed free abelian monoid ``F``.
EXAMPLES::
sage: F = FreeMonoid(index_set=tuple('abcde'))
sage: x = F( [(1, 2), (0, 1), (3, 2), (0, 1)] )
sage: y = F( ((1, 2), (0, 1), [3, 2], [0, 1]) )
sage: z = F( reversed([(0, 1), (3, 2), (0, 1), (1, 2)]) )
sage: x == y and y == z
True
sage: TestSuite(x).run()
"""
IndexedMonoidElement.__init__(self, F, tuple(map(tuple, x)))
def __hash__(self):
r"""
TESTS::
sage: F = FreeMonoid(index_set=tuple('abcde'))
sage: hash(F ([(1,2),(0,1)]) )
2401565693828035651 # 64-bit
1164080195 # 32-bit
sage: hash(F ([(0,2),(1,1)]) )
-3359280905493236379 # 64-bit
-1890405019 # 32-bit
"""
return hash(self._monomial)
def _sorted_items(self):
"""
Return the sorted items (i.e factors) of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: x = a*b^2*e*d
sage: x._sorted_items()
((0, 1), (1, 2), (4, 1), (3, 1))
sage: F.print_options(generator_cmp = lambda x,y: -cmp(x,y))
sage: x._sorted_items()
((0, 1), (1, 2), (4, 1), (3, 1))
sage: F.print_options(generator_cmp=cmp) # reset to original state
.. SEEALSO::
:meth:`_repr_`, :meth:`_latex_`, :meth:`print_options`
"""
return self._monomial
def _mul_(self, other):
"""
Multiply ``self`` by ``other``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a*b^2*e*d
F[0]*F[1]^2*F[4]*F[3]
sage: (a*b^2*d^2) * (d^4*b*e)
F[0]*F[1]^2*F[3]^6*F[1]*F[4]
"""
if not self._monomial:
return other
if not other._monomial:
return self
ret = list(self._monomial)
rhs = list(other._monomial)
if ret[-1][0] == rhs[0][0]:
rhs[0] = (rhs[0][0], rhs[0][1] + ret.pop()[1])
ret += rhs
return self.__class__(self.parent(), tuple(ret))
def __len__(self):
"""
Return the length of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: elt = a*c^3*b^2*a
sage: elt.length()
7
sage: len(elt)
7
"""
return sum(exp for gen,exp in self._monomial)
length = __len__
class IndexedFreeAbelianMonoidElement(IndexedMonoidElement):
"""
An element of an indexed free abelian monoid.
"""
def __init__(self, F, x):
"""
Create the element ``x`` of an indexed free abelian monoid ``F``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: x = F([(0, 1), (2, 2), (-1, 2)])
sage: y = F({0:1, 2:2, -1:2})
sage: z = F(reversed([(0, 1), (2, 2), (-1, 2)]))
sage: x == y and y == z
True
sage: TestSuite(x).run()
"""
IndexedMonoidElement.__init__(self, F, dict(x))
def _sorted_items(self):
"""
Return the sorted items (i.e factors) of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: x = a*b^2*e*d
sage: x._sorted_items()
[(0, 1), (1, 2), (3, 1), (4, 1)]
sage: F.print_options(generator_cmp = lambda x,y: -cmp(x,y))
sage: x._sorted_items()
[(4, 1), (3, 1), (1, 2), (0, 1)]
sage: F.print_options(generator_cmp=cmp) # reset to original state
.. SEEALSO::
:meth:`_repr_`, :meth:`_latex_`, :meth:`print_options`
"""
print_options = self.parent().print_options()
v = self._monomial.items()
try:
v.sort(cmp = print_options['generator_cmp'])
except Exception: # Sorting the output is a plus, but if we can't, no big deal
pass
return v
def __hash__(self):
r"""
TESTS::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: hash( F([(0,1), (2,2)]) )
8087055352805725849 # 64-bit
250091161 # 32-bit
sage: hash( F([(2,1)]) )
5118585357534560720 # 64-bit
1683816912 # 32-bit
"""
return hash(frozenset(self._monomial.items()))
def _mul_(self, other):
"""
Multiply ``self`` by ``other``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a*b^2*e*d
F[0]*F[1]^2*F[3]*F[4]
"""
return self.__class__(self.parent(),
dict_addition([self._monomial, other._monomial]))
def __pow__(self, n):
"""
Raise ``self`` to the power of ``n``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: x = a*b^2*e*d; x
F[0]*F[1]^2*F[3]*F[4]
sage: x^3
F[0]^3*F[1]^6*F[3]^3*F[4]^3
sage: x^0
1
"""
if not isinstance(n, (int, long, Integer)):
raise TypeError("Argument n (= {}) must be an integer".format(n))
if n < 0:
raise ValueError("Argument n (= {}) must be positive".format(n))
if n == 1:
return self
if n == 0:
return self.parent().one()
return self.__class__(self.parent(), {k:v*n for k,v in self._monomial.iteritems()})
def __floordiv__(self, elt):
"""
Cancel the element ``elt`` out of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: elt = a*b*c^3*d^2; elt
F[0]*F[1]*F[2]^3*F[3]^2
sage: elt // a
F[1]*F[2]^3*F[3]^2
sage: elt // c
F[0]*F[1]*F[2]^2*F[3]^2
sage: elt // (a*b*d^2)
F[2]^3
sage: elt // a^4
Traceback (most recent call last):
...
ValueError: invalid cancellation
"""
d = copy(self._monomial)
for k,v in elt._monomial.iteritems():
d[k] -= v
for k,v in d.items():
if v < 0:
raise ValueError("invalid cancellation")
if v == 0:
del d[k]
return self.__class__(self.parent(), d)
def __len__(self):
"""
Return the length of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: elt = a*c^3*b^2*a
sage: elt.length()
7
sage: len(elt)
7
"""
m = self._monomial
return sum(m[gen] for gen in m)
length = __len__
def dict(self):
"""
Return ``self`` as a dictionary.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: (a*c^3).dict()
{0: 1, 2: 3}
"""
return copy(self._monomial)
class IndexedMonoid(Parent, IndexedGenerators, UniqueRepresentation):
"""
Base class for monoids with an indexed set of generators.
INPUT:
- ``indices`` -- the indices for the generators
For the optional arguments that control the printing, see
:class:`~sage.structure.indexed_generators.IndexedGenerators`.
"""
@staticmethod
def __classcall__(cls, indices, prefix=None, names=None, **kwds):
"""
TESTS::
sage: F = FreeAbelianMonoid(index_set=['a','b','c'])
sage: G = FreeAbelianMonoid(index_set=('a','b','c'))
sage: H = FreeAbelianMonoid(index_set=tuple('abc'))
sage: F is G and F is H
True
sage: F = FreeAbelianMonoid(index_set=['a','b','c'], latex_bracket=['LEFT', 'RIGHT'])
sage: F.print_options()['latex_bracket']
('LEFT', 'RIGHT')
sage: F is G
False
sage: Groups.Commutative.free()
Traceback (most recent call last):
...
ValueError: either the indices or names must be given
"""
indices, names, prefix = parse_indices_names(indices, names, prefix, kwds)
if prefix is None:
prefix = "F"
# bracket or latex_bracket might be lists, so convert
# them to tuples so that they're hashable.
bracket = kwds.get('bracket', None)
if isinstance(bracket, list):
kwds['bracket'] = tuple(bracket)
latex_bracket = kwds.get('latex_bracket', None)
if isinstance(latex_bracket, list):
kwds['latex_bracket'] = tuple(latex_bracket)
return super(IndexedMonoid, cls).__classcall__(cls, indices, prefix, names=names, **kwds)
def __init__(self, indices, prefix, category=None, names=None, **kwds):
"""
Initialize ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: TestSuite(F).run()
sage: F = FreeMonoid(index_set=tuple('abcde'))
sage: TestSuite(F).run()
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: TestSuite(F).run()
sage: F = FreeAbelianMonoid(index_set=tuple('abcde'))
sage: TestSuite(F).run()
"""
self._indices = indices
category = Monoids().or_subcategory(category)
if indices.cardinality() == 0:
category = category.Finite()
else:
category = category.Infinite()
if indices in Sets().Finite():
category = category.FinitelyGeneratedAsMagma()
Parent.__init__(self, names=names, category=category)
# ignore the optional 'key' since it only affects CachedRepresentation
kwds.pop('key', None)
IndexedGenerators.__init__(self, indices, prefix, **kwds)
def _first_ngens(self, n):
"""
Used by the preparser for ``F.<x> = ...``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: F._first_ngens(3)
(F[0], F[1], F[-1])
"""
it = iter(self._indices)
return tuple(self.gen(next(it)) for i in range(n))
def _element_constructor_(self, x=None):
"""
Create an element of this abelian monoid from ``x``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F(F.gen(2))
F[2]
sage: F([[1, 3], [-2, 12]])
F[-2]^12*F[1]^3
sage: F(-5)
Traceback (most recent call last):
...
TypeError: unable to convert -5, use gen() instead
"""
if x is None:
return self.one()
if x in self._indices:
raise TypeError("unable to convert {!r}, use gen() instead".format(x))
return self.element_class(self, x)
def _an_element_(self):
"""
Return an element of ``self``.
EXAMPLES::
sage: G = FreeAbelianMonoid(index_set=ZZ)
sage: G.an_element()
F[-1]^3*F[0]*F[1]^3
sage: G = FreeMonoid(index_set=tuple('ab'))
sage: G.an_element()
F['a']^2*F['b']^2
"""
x = self.one()
I = self._indices
try:
x *= self.gen(I.an_element())
except Exception:
pass
try:
g = iter(self._indices)
for c in range(1,4):
x *= self.gen(next(g)) ** c
except Exception:
pass
return x
def cardinality(self):
r"""
Return the cardinality of ``self``, which is `\infty` unless this is
the trivial monoid.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: F.cardinality()
+Infinity
sage: F = FreeMonoid(index_set=())
sage: F.cardinality()
1
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F.cardinality()
+Infinity
sage: F = FreeAbelianMonoid(index_set=())
sage: F.cardinality()
1
"""
if self._indices.cardinality() == 0:
return ZZ.one()
return infinity
@cached_method
def monoid_generators(self):
"""
Return the monoid generators of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F.monoid_generators()
Lazy family (Generator map from Integer Ring to
Free abelian monoid indexed by Integer Ring(i))_{i in Integer Ring}
sage: F = FreeAbelianMonoid(index_set=tuple('abcde'))
sage: sorted(F.monoid_generators())
[F['a'], F['b'], F['c'], F['d'], F['e']]
"""
if self._indices.cardinality() == infinity:
gen = PoorManMap(self.gen, domain=self._indices, codomain=self, name="Generator map")
return Family(self._indices, gen)
return Family(self._indices, self.gen)
gens = monoid_generators
class IndexedFreeMonoid(IndexedMonoid):
"""
Free monoid with an indexed set of generators.
INPUT:
- ``indices`` -- the indices for the generators
For the optional arguments that control the printing, see
:class:`~sage.structure.indexed_generators.IndexedGenerators`.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: F.gen(15)^3 * F.gen(2) * F.gen(15)
F[15]^3*F[2]*F[15]
sage: F.gen(1)
F[1]
Now we examine some of the printing options::
sage: F = FreeMonoid(index_set=ZZ, prefix='X', bracket=['|','>'])
sage: F.gen(2) * F.gen(12)
X|2>*X|12>
"""
def _repr_(self):
"""
Return a string representation of ``self``.
EXAMPLES::
sage: FreeMonoid(index_set=ZZ)
Free monoid indexed by Integer Ring
"""
return "Free monoid indexed by {}".format(self._indices)
Element = IndexedFreeMonoidElement
@cached_method
def one(self):
"""
Return the identity element of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: F.one()
1
"""
return self.element_class(self, ())
def gen(self, x):
"""
The generator indexed by ``x`` of ``self``.
EXAMPLES::
sage: F = FreeMonoid(index_set=ZZ)
sage: F.gen(0)
F[0]
sage: F.gen(2)
F[2]
"""
if x not in self._indices:
raise IndexError("{} is not in the index set".format(x))
try:
return self.element_class(self, ((self._indices(x),1),))
except TypeError: # Backup (if it is a string)
return self.element_class(self, ((x,1),))
class IndexedFreeAbelianMonoid(IndexedMonoid):
"""
Free abelian monoid with an indexed set of generators.
INPUT:
- ``indices`` -- the indices for the generators
For the optional arguments that control the printing, see
:class:`~sage.structure.indexed_generators.IndexedGenerators`.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F.gen(15)^3 * F.gen(2) * F.gen(15)
F[2]*F[15]^4
sage: F.gen(1)
F[1]
Now we examine some of the printing options::
sage: F = FreeAbelianMonoid(index_set=Partitions(), prefix='A', bracket=False, scalar_mult='%')
sage: F.gen([3,1,1]) * F.gen([2,2])
A[2, 2]%A[3, 1, 1]
"""
def _repr_(self):
"""
Return a string representation of ``self``.
EXAMPLES::
sage: FreeAbelianMonoid(index_set=ZZ)
Free abelian monoid indexed by Integer Ring
"""
return "Free abelian monoid indexed by {}".format(self._indices)
def _element_constructor_(self, x=None):
"""
Create an element of ``self`` from ``x``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F(F.gen(2))
F[2]
sage: F([[1, 3], [-2, 12]])
F[-2]^12*F[1]^3
sage: F({1:3, -2: 12})
F[-2]^12*F[1]^3
"""
if isinstance(x, (list, tuple)):
x = dict(x)
return IndexedMonoid._element_constructor_(self, x)
Element = IndexedFreeAbelianMonoidElement
@cached_method
def one(self):
"""
Return the identity element of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F.one()
1
"""
return self.element_class(self, {})
def gen(self, x):
"""
The generator indexed by ``x`` of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: F.gen(0)
F[0]
sage: F.gen(2)
F[2]
"""
if x not in self._indices:
raise IndexError("{} is not in the index set".format(x))
try:
return self.element_class(self, {self._indices(x):1})
except TypeError: # Backup (if it is a string)
return self.element_class(self, {x:1})