This repository has been archived by the owner on Jan 30, 2023.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
/
libgap_wrapper.pyx
652 lines (503 loc) · 17 KB
/
libgap_wrapper.pyx
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
"""
LibGAP-based Groups
This module provides helper class for wrapping GAP groups via
:mod:`~sage.libs.gap.libgap`. See :mod:`~sage.groups.free_group` for an
example how they are used.
The parent class keeps track of the libGAP element object, to use it
in your Python parent you have to derive both from the suitable group
parent and :class:`ParentLibGAP` ::
sage: from sage.groups.libgap_wrapper import ElementLibGAP, ParentLibGAP
sage: from sage.groups.group import Group
sage: class FooElement(ElementLibGAP):
... pass
sage: class FooGroup(Group, ParentLibGAP):
... Element = FooElement
... def __init__(self):
... lg = libgap(libgap.CyclicGroup(3)) # dummy
... ParentLibGAP.__init__(self, lg)
... Group.__init__(self)
Note how we call the constructor of both superclasses to initialize
``Group`` and ``ParentLibGAP`` separately. The parent class implements
its output via LibGAP::
sage: FooGroup()
<pc group of size 3 with 1 generators>
sage: type(FooGroup().gap())
<type 'sage.libs.gap.element.GapElement'>
The element class is a subclass of
:class:`~sage.structure.element.MultiplicativeGroupElement`. To use
it, you just inherit from :class:`ElementLibGAP` ::
sage: element = FooGroup().an_element()
sage: element
f1
The element class implements group operations and printing via LibGAP::
sage: element._repr_()
'f1'
sage: element * element
f1^2
AUTHORS:
- Volker Braun
"""
##############################################################################
# Copyright (C) 2012 Volker Braun <vbraun.name@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# The full text of the GPL is available at:
#
# http://www.gnu.org/licenses/
##############################################################################
from sage.libs.gap.element cimport GapElement
from sage.rings.integer import Integer
from sage.rings.integer_ring import IntegerRing
from sage.misc.cachefunc import cached_method
from sage.structure.sage_object import SageObject
from sage.structure.element cimport Element
class ParentLibGAP(SageObject):
"""
A class for parents to keep track of the GAP parent.
This is not a complete group in Sage, this class is only a base
class that you can use to implement your own groups with
LibGAP. See :mod:`~sage.groups.libgap_group` for a minimal example
of a group that is actually usable.
Your implementation definitely needs to supply
* ``__reduce__()``: serialize the LibGAP group. Since GAP does not
support Python pickles natively, you need to figure out yourself
how you can recreate the group from a pickle.
INPUT:
- ``libgap_parent`` -- the libgap element that is the parent in
GAP.
- ``ambient`` -- A derived class of :class:`ParentLibGAP` or
``None`` (default). The ambient class if ``libgap_parent`` has
been defined as a subgroup.
EXAMPLES::
sage: from sage.groups.libgap_wrapper import ElementLibGAP, ParentLibGAP
sage: from sage.groups.group import Group
sage: class FooElement(ElementLibGAP):
... pass
sage: class FooGroup(Group, ParentLibGAP):
... Element = FooElement
... def __init__(self):
... lg = libgap(libgap.CyclicGroup(3)) # dummy
... ParentLibGAP.__init__(self, lg)
... Group.__init__(self)
sage: FooGroup()
<pc group of size 3 with 1 generators>
"""
def __init__(self, libgap_parent, ambient=None):
"""
The Python constructor.
TESTS::
sage: G = FreeGroup(3)
sage: TestSuite(G).run()
"""
assert isinstance(libgap_parent, GapElement)
self._libgap = libgap_parent
self._ambient = ambient
def ambient(self):
"""
Return the ambient group of a subgroup.
OUTPUT:
A group containing ``self``. If ``self`` has not been defined
as a subgroup, we just return ``self``.
EXAMPLES::
sage: G = FreeGroup(3)
sage: G.ambient() is G
True
"""
if self._ambient is None:
return self
else:
return self._ambient
def is_subgroup(self):
"""
Return whether the group was defined as a subgroup of a bigger
group.
You can access the contaning group with :meth:`ambient`.
OUTPUT:
Boolean.
EXAMPLES::
sage: G = FreeGroup(3)
sage: G.is_subgroup()
False
"""
return self._ambient is not None
def _subgroup_constructor(self, libgap_subgroup):
"""
Return the class of a subgroup.
You should override this with a derived class. Its constructor
must accept the same arguments as :meth:`__init__`.
OUTPUT:
A new instance of a group (derived class of
:class:`ParentLibGAP`).
TESTS::
sage: F.<a,b> = FreeGroup()
sage: G = F.subgroup([a^2*b]); G
Group([ a^2*b ])
sage: F._subgroup_constructor(G.gap())._repr_()
'Group([ a^2*b ])'
"""
from sage.groups.libgap_group import GroupLibGAP
return GroupLibGAP(libgap_subgroup, ambient=self)
def subgroup(self, generators):
"""
Return the subgroup generated.
INPUT:
- ``generators`` -- a list/tuple/iterable of group elements.
OUTPUT:
The subgroup generated by ``generators``.
EXAMPLES::
sage: F.<a,b> = FreeGroup()
sage: G = F.subgroup([a^2*b]); G
Group([ a^2*b ])
sage: G.gens()
(a^2*b,)
"""
generators = [ g if isinstance(g, GapElement) else g.gap()
for g in generators ]
G = self.gap()
H = G.Subgroup(generators)
return self._subgroup_constructor(H)
def gap(self):
"""
Returns the gap representation of self
OUTPUT:
A :class:`~sage.libs.gap.element.GapElement`
EXAMPLES::
sage: G = FreeGroup(3); G
Free Group on generators {x0, x1, x2}
sage: G.gap()
<free group on the generators [ x0, x1, x2 ]>
sage: G.gap().parent()
C library interface to GAP
sage: type(G.gap())
<type 'sage.libs.gap.element.GapElement'>
This can be useful, for example, to call GAP functions that
are not wrapped in Sage::
sage: G = FreeGroup(3)
sage: H = G.gap()
sage: H.DirectProduct(H)
<fp group on the generators [ f1, f2, f3, f4, f5, f6 ]>
sage: H.DirectProduct(H).RelatorsOfFpGroup()
[ f1^-1*f4^-1*f1*f4, f1^-1*f5^-1*f1*f5, f1^-1*f6^-1*f1*f6, f2^-1*f4^-1*f2*f4,
f2^-1*f5^-1*f2*f5, f2^-1*f6^-1*f2*f6, f3^-1*f4^-1*f3*f4, f3^-1*f5^-1*f3*f5,
f3^-1*f6^-1*f3*f6 ]
We can also convert directly to libgap::
sage: libgap(GL(2, ZZ))
GL(2,Integers)
"""
return self._libgap
_libgap_ = _gap_ = gap
@cached_method
def _gap_gens(self):
"""
Return the generators as a LibGAP object
OUTPUT:
A :class:`~sage.libs.gap.element.GapElement`
EXAMPLES:
sage: G = FreeGroup(2)
sage: G._gap_gens()
[ x0, x1 ]
sage: type(_)
<type 'sage.libs.gap.element.GapElement_List'>
"""
return self._libgap.GeneratorsOfGroup()
@cached_method
def ngens(self):
"""
Return the number of generators of self.
OUTPUT:
Integer.
EXAMPLES::
sage: G = FreeGroup(2)
sage: G.ngens()
2
TESTS::
sage: type(G.ngens())
<type 'sage.rings.integer.Integer'>
"""
return self._gap_gens().Length().sage()
def _repr_(self):
"""
Return a string representation
OUTPUT:
String.
TESTS::
sage: from sage.groups.libgap_wrapper import ElementLibGAP, ParentLibGAP
sage: G.<a,b> =FreeGroup()
sage: ParentLibGAP._repr_(G)
'<free group on the generators [ a, b ]>'
"""
return self._libgap._repr_()
def gen(self, i):
"""
Return the `i`-th generator of self.
.. warning::
Indexing starts at `0` as usual in Sage/Python. Not as in
GAP, where indexing starts at `1`.
INPUT:
- ``i`` -- integer between `0` (inclusive) and :meth:`ngens`
(exclusive). The index of the generator.
OUTPUT:
The `i`-th generator of the group.
EXAMPLES::
sage: G = FreeGroup('a, b')
sage: G.gen(0)
a
sage: G.gen(1)
b
"""
if not (0 <= i < self.ngens()):
raise ValueError('i must be in range(ngens)')
gap = self._gap_gens()[i]
return self.element_class(self, gap)
@cached_method
def gens(self):
"""
Returns the generators of the group.
EXAMPLES::
sage: G = FreeGroup(2)
sage: G.gens()
(x0, x1)
sage: H = FreeGroup('a, b, c')
sage: H.gens()
(a, b, c)
:meth:`generators` is an alias for :meth:`gens` ::
sage: G = FreeGroup('a, b')
sage: G.generators()
(a, b)
sage: H = FreeGroup(3, 'x')
sage: H.generators()
(x0, x1, x2)
"""
return tuple( self.gen(i) for i in range(self.ngens()) )
generators = gens
@cached_method
def one(self):
"""
Returns the identity element of self
EXAMPLES::
sage: G = FreeGroup(3)
sage: G.one()
1
sage: G.one() == G([])
True
sage: G.one().Tietze()
()
"""
return self.element_class(self, self.gap().Identity())
def _an_element_(self):
"""
Returns an element of self.
EXAMPLES::
sage: G.<a,b> = FreeGroup()
sage: G._an_element_()
a*b
"""
from sage.misc.all import prod
return prod(self.gens())
cdef class ElementLibGAP(MultiplicativeGroupElement):
"""
A class for LibGAP-based Sage group elements
INPUT:
- ``parent`` -- the Sage parent
- ``libgap_element`` -- the libgap element that is being wrapped
EXAMPLES::
sage: from sage.groups.libgap_wrapper import ElementLibGAP, ParentLibGAP
sage: from sage.groups.group import Group
sage: class FooElement(ElementLibGAP):
... pass
sage: class FooGroup(Group, ParentLibGAP):
... Element = FooElement
... def __init__(self):
... lg = libgap(libgap.CyclicGroup(3)) # dummy
... ParentLibGAP.__init__(self, lg)
... Group.__init__(self)
sage: FooGroup()
<pc group of size 3 with 1 generators>
sage: FooGroup().gens()
(f1,)
"""
def __init__(self, parent, libgap_element):
"""
The Python constructor
TESTS::
sage: G = FreeGroup(2)
sage: g = G.an_element()
sage: TestSuite(g).run()
"""
MultiplicativeGroupElement.__init__(self, parent)
assert isinstance(parent, ParentLibGAP)
if isinstance(libgap_element, GapElement):
self._libgap = libgap_element
else:
if libgap_element == 1:
self._libgap = self.parent().gap().Identity()
else:
raise TypeError('need a libgap group element or "1" in constructor')
cpdef GapElement gap(self):
"""
Returns a LibGAP representation of the element
OUTPUT:
A :class:`~sage.libs.gap.element.GapElement`
EXAMPLES::
sage: G.<a,b> = FreeGroup('a, b')
sage: x = G([1, 2, -1, -2])
sage: x
a*b*a^-1*b^-1
sage: xg = x.gap()
sage: xg
a*b*a^-1*b^-1
sage: type(xg)
<type 'sage.libs.gap.element.GapElement'>
"""
return self._libgap
_gap_ = gap
def is_one(self):
"""
Test whether the group element is the trivial element.
OUTPUT:
Boolean.
EXAMPLES::
sage: G.<a,b> = FreeGroup('a, b')
sage: x = G([1, 2, -1, -2])
sage: x.is_one()
False
sage: (x * ~x).is_one()
True
"""
return self == self.parent().one()
def _repr_(self):
"""
Return a string representation.
OUTPUT:
String.
EXAMPLES::
sage: G.<a,b> = FreeGroup()
sage: a._repr_()
'a'
sage: type(a)
<class 'sage.groups.free_group.FreeGroup_class_with_category.element_class'>
sage: x = G([1, 2, -1, -2])
sage: x._repr_()
'a*b*a^-1*b^-1'
sage: y = G([2, 2, 2, 1, -2, -2, -2])
sage: y._repr_()
'b^3*a*b^-3'
sage: G.one()
1
"""
if self.is_one():
return '1'
else:
return self._libgap._repr_()
def _latex_(self):
r"""
Return a LaTeX representation
OUTPUT:
String. A valid LaTeX math command sequence.
EXAMPLES::
sage: from sage.groups.libgap_group import GroupLibGAP
sage: G = GroupLibGAP(libgap.FreeGroup('a', 'b'))
sage: g = G.gen(0) * G.gen(1)
sage: g._latex_()
"ab%\n"
"""
try:
return self.gap().LaTeX()
except ValueError:
from sage.misc.latex import latex
return latex(self._repr_())
cpdef _mul_(left, MonoidElement right):
"""
Multiplication of group elements
TESTS::
sage: G = FreeGroup('a, b')
sage: x = G([1, 2, -1, -2])
sage: y = G([2, 2, 2, 1, -2, -2, -2])
sage: x*y # indirect doctest
a*b*a^-1*b^2*a*b^-3
sage: y*x # indirect doctest
b^3*a*b^-3*a*b*a^-1*b^-1
sage: x*y == x._mul_(y)
True
sage: y*x == y._mul_(x)
True
"""
P = left.parent()
return P.element_class(P, left.gap() * right.gap())
cpdef int _cmp_(left, Element right) except -2:
"""
This method implements comparison.
TESTS::
sage: G.<a,b> = FreeGroup('a, b')
sage: G_gap = G.gap()
sage: G_gap == G_gap # indirect doctest
True
sage: cmp(G.gap(), G.gap()) # indirect doctest
0
sage: x = G([1, 2, -1, -2])
sage: y = G([2, 2, 2, 1, -2, -2, -2])
sage: x == x*y*y^(-1) # indirect doctest
True
sage: cmp(x,y)
-1
sage: x < y
True
"""
return cmp((<ElementLibGAP>left)._libgap,
(<ElementLibGAP>right)._libgap)
cpdef _div_(left, MultiplicativeGroupElement right):
"""
Division of group elements.
TESTS::
sage: G = FreeGroup('a, b')
sage: x = G([1, 2, -1, -2])
sage: y = G([2, 2, 2, 1, -2, -2, -2])
sage: x/y # indirect doctest
a*b*a^-1*b^2*a^-1*b^-3
sage: y/x # indirect doctest
b^3*a*b^-2*a*b^-1*a^-1
sage: x/y == x.__div__(y)
True
sage: x/y == y.__div__(x)
False
"""
P = left.parent()
return P.element_class(P, left.gap() / right.gap())
def __pow__(self, n, dummy):
"""
Implement exponentiation.
TESTS::
sage: G = FreeGroup('a, b')
sage: x = G([1, 2, -1, -2])
sage: y = G([2, 2, 2, 1, -2, -2, -2])
sage: y^(2) # indirect doctest
b^3*a^2*b^-3
sage: x^(-3) # indirect doctest
(b*a*b^-1*a^-1)^3
sage: y^3 == y.__pow__(3)
True
"""
if n not in IntegerRing():
raise TypeError("exponent must be an integer")
P = self.parent()
return P.element_class(P, self.gap() ** n)
def __invert__(self):
"""
Return the inverse of self.
TESTS::
sage: G = FreeGroup('a, b')
sage: x = G([1, 2, -1, -2])
sage: y = G([2, 2, 2, 1, -2, -2, -2])
sage: x.__invert__()
b*a*b^-1*a^-1
sage: y.__invert__()
b^3*a^-1*b^-3
sage: ~x
b*a*b^-1*a^-1
sage: x.inverse()
b*a*b^-1*a^-1
"""
P = self.parent()
return P.element_class(P, self.gap().Inverse())
inverse = __invert__