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free_group.py
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free_group.py
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r"""
Free group of finite rank.
"""
#*****************************************************************************
# Copyright (C) 2013 Thierry Coulbois <thierry.coulbois@univ-amu.fr>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.combinat.words.alphabet import build_alphabet
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.parent import Parent
from sage.groups.group import Group
from sage.categories.groups import Groups
from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
from inverse_alphabet import build_alphabet_with_inverses
from free_group_word import FreeGroupWord
class FreeGroup(UniqueRepresentation, Group):
"""
Free group of finite rank.
EXAMPLES::
sage: A=AlphabetWithInverses(['a','b'])
sage: FreeGroup(A)
Free group over ['a', 'b']
sage: FreeGroup(3)
Free group over ['a', 'b', 'c']
sage: A=AlphabetWithInverses(2,type='x0')
sage: FreeGroup(A)
Free group over ['x0', 'x1']
AUTHORS:
- Thierry Coulbois (2013-05-16): beta.0 version
"""
Element = FreeGroupWord
@staticmethod
def __classcall__(cls, *data, **kwds):
pos, neg = build_alphabet_with_inverses(*data, **kwds)
return super(FreeGroup, cls).__classcall__(cls, pos, neg)
def __init__(self, pos, neg=None):
r"""
INPUT:
- ``pos`` - the alphabet of positive letters
- ``neg`` - the alphabet of negative letters
- ``bij`` - bijection between positive and negative letters
"""
Group.__init__(self, category=(Groups(),InfiniteEnumeratedSets()))
self._pos = pos
self._neg = neg
self._invert = {}
self._invert.update(zip(pos,neg))
self._invert.update(zip(neg,pos))
self._alphabet = build_alphabet(pos.list() + neg.list())
def gens(self):
r"""
Return the canonical generating set for this free group.
"""
return tuple(self([letter],check=False) for letter in self.positive_letters())
def __iter__(self):
r"""
TESTS::
sage: it = iter(FreeGroup('ab'))
sage: [it.next() for _ in xrange(25)]
[THE_EMPTY_WORD,
a,
b,
A,
B,
aa,
...
aBa,
aBA]
sage: it.next()
aBB
"""
n = 0
while True:
for w in self.subset(n): yield w
n += 1
def inverse_letter(self, a):
r"""
Return the inverse of the letter ``a`` or raise a ValueError.
EXAMPLES::
sage: F = FreeGroup('abc','XYZ')
sage: F.inverse_letter('b')
'Y'
sage: F.inverse_letter('Z')
'c'
"""
try:
return self._invert[a]
except StandardError:
raise ValueError("the letter %s is not in the alphabet"%a)
def one(self):
r"""
Return the identity element in self.
EXAMPLES::
sage: F = FreeGroup('abc')
sage: F.one()
THE_EMPTY_WORD
"""
return self([])
empty_word = one
def alphabet(self):
r"""
Return the set of words of length 1.
EXAMPLES::
sage: F = FreeGroup('abc')
sage: F.alphabet()
{'a', 'b', 'c', 'A', 'B', 'C'}
"""
return self._alphabet
def positive_letters(self):
r"""
Return the set of positive letters
EXAMPLES::
sage: F = FreeGroup('abc')
sage: F.positive_letters()
{'a', 'b', 'c'}
"""
return self._pos
def negative_letters(self):
r"""
Return the set of negative letters
EXAMPLES::
sage: F = FreeGroup('abc')
sage: F.negative_letters()
{'A', 'B', 'C'}
"""
return self._neg
def subset(self, n):
r"""
Return the set of elements of given length.
EXAMPLES::
sage: F = FreeGroup('ab')
sage: F.subset(2)
Words of length 4 in Free group over {'a', 'b'}
"""
try:
n = n.__index__()
except StandardError:
raise TypeError("n should be integer")
return FreeGroup_n(self, n)
def _repr_(self):
"""
String representation for free group
TESTS::
sage: FreeGroup('ab') # indirect doctest
Free group over {'a', 'b'}
"""
return "Free group over %s"%self.positive_letters()
def _element_constructor_(self, data=None, check=True):
r"""
Build an element of the free group,
If ``check`` is True, then check that the validity of letters and that
the word is reduced.
TESTS::
sage: F = FreeGroup('ab')
sage: F('abAb') # indirect doctest
abAb
sage: F('abAaBbBA') # indirect doctest
THE_EMPTY_WORD
sage: F('aaac') # indirect doctest
Traceback (most recent call last):
...
ValueError: the letter c is not in the alphabet
"""
if data is None:
data=[]
return self.element_class(self, data, check)
#TODO
def identity_automorphism(self):
"""
Identity automorphism of ``self``.
"""
morph=dict((a,self([a])) for a in self.positive_letters())
return FreeGroupAutomorphism(morph,group=self)
#TODO
def dehn_twist(self,a,b,on_left=False):
"""
Dehn twist automorphism of ``self``.
if ``on_left`` is ``False``: ``a -> ab``
if ``on_left`` is ``True``: ``a -> ba``
EXAMPLES
sage: F=FreeGroup(3)
sage: F.dehnt_twist('a','c')
a->ac, b->b, c->c
sage: F.dehn_twist('A','c')
a->Ca,b->b,c->c
"""
A = self.alphabet()
if a not in A:
raise ValueError, "Letter %s not in alphabet" %str(a)
if b not in A:
raise ValueError, "Letter %s not in alphabet" %str(b)
if a == b:
raise ValueError, "Letter a=%s should be different from b=%s" %(str(a),str(b))
if A.are_inverse(a,b):
raise ValueError, "Letter a=%s should be different from the inverse of b=%s" %(str(a),str(b))
morphism = dict((letter,self([letter])) for letter in self.positive_letters())
if A.is_positive_letter(a):
if on_left:
morphism[a] = self([b,a])
else:
morphism[a] = self([a,b])
else:
a = A.inverse_letter(a)
b = A.inverse_letter(b)
if on_left:
morphism[a] = self([a,b])
else:
morphism[a] = self([b,a])
return FreeGroupAutomorphism(morphism,group=self)
#TODO
def random_automorphism(self,length=1):
"""
Random automorphism of ``self``.
The output is a random word of given ``length`` on the set of Dehn twists.
"""
if length==0: return self.identity_automorphism()
A=self.alphabet()
a=A.random_letter()
b=A.random_letter([a,A.inverse_letter(a)])
result=self.dehn_twist(a,b)
for i in xrange(length-1):
new_a=A.random_letter()
if new_a==a:
b=A.random_letter([a,A.inverse_letter(a),A.inverse_letter(b)])
else:
a=new_a
b=A.random_letter([a,A.inverse_letter(a)])
result=result*self.dehn_twist(a,b)
return result
#TODO
def _surface_dehn_twist_e(self,i):
a=self._alphabet[2*i]
b=self._alphabet[2*i+1]
return self.dehn_twist(a,b,True)
#TODO
def _surface_dehn_twist_c(self,i):
A=self._alphabet
result=dict((a,self([a])) for a in A.positive_letters())
result[A[2*i+1]]=self([A[2*i+2],A.inverse_letter(A[2*i]),A[2*i+1]])
result[A[2*i+3]]=self([A[2*i+3],A[2*i],A.inverse_letter(A[2*i+2])])
return FreeGroupAutomorphism(result,group=self)
#TODO
def _surface_dehn_twist_m(self,i):
A=self._alphabet
result={}
for j in xrange(2*i+1):
result[A[j]]=self([A[j]])
a=A[2*i]
result[A[2*i+1]]=self([a,A[2*i+1]])
aa=A.inverse_letter(a)
for j in xrange(2*i+2,len(A)):
result[A[j]]=self([a,A[j],aa])
return FreeGroupAutomorphism(result,group=self)
#TODO
def surface_dehn_twist(self,k):
"""
Dehn twist of the surface (with one boundary component) with
fundamental group ``self``.
The surface is assumed to have genus g and 1 boundary
component. The fundamental group has rank 2g, thus ``self`` is
assumed to be of even rank.
``k`` is an integer 0<=k<3g-1.
MCG(S_{g,1}) is generated by the Dehn twist along
the curves:
- g equators e_i,
- g meridian m_i
- g-1 circles c_i around two consecutive 'holes'.
for 0<=k<g returns the Dehn twist along e_i with i=k
for g<=k<2g returns the Dehn twist along m_i with i=k-g
for 2g<=k<3g-1 returns the Dehn twist along c_i with i=k-2g
The fundamental group has 2g generators. We fix the base point
on the boundary. The generators are:
- g x_i that turns around the i-th hole
- g y_i that goes inside the i-th hole
T_{e_i}: x_j-> x_j, x_i->y_ix_i, y_j->y_j
T_{m_i}: x_j->x_j, y_j->y_j, j<i
x_i->x_i, y_i->x_iy_i
x_j->x_ix_jx_i\inv, y_j->x_iy_jx_i\inv
T_{c_i}: x_j->x_j, y_j->y_j, y_i->x_{i+1}x_i\inv y_i, y_{i+1}->y_{i+1}x_{i+1}x_i\inv
WARNING:
``self`` is assumed to be of even rank.
"""
assert len(self._alphabet)%2==0
g=len(self._alphabet)/2
if (0<=k and k<g): result=self._surface_dehn_twist_e(k)
elif (g<=k and k<2*g): result=self._surface_dehn_twist_m(k-g)
elif (2*g<=k and k<3*g-1): result=self._surface_dehn_twist_c(k-2*g)
return result
#TODO
def random_mapping_class(self,n=1):
"""
Random mapping class of length (as a product of
generating dehn twists) at most ``n``. `
WARNING:
The rank of ``self` is assumed to be even.
"""
assert len(self._alphabet)%2==0
if n==0:
return self.identity_automorphism()
r=3*len(self._alphabet)/2-2
i=randint(0,r)
j=randint(0,1)
if j==0:
result=self.surface_dehn_twist(i)
else:
result=self.surface_dehn_twist(i).inverse()
for ii in xrange(n-1):
l=randint(0,1)
if j==l:
i=randint(0,r)
else:
k=randint(0,r-1)
if k>=i: i=k+1
j=l
if j==0:
result=result*self.surface_dehn_twist(i)
else:
result=result*self.surface_dehn_twist(i).inverse()
return result
#TODO
def braid_automorphism(self,i,inverse=False):
"""
Automorphism of ``self`` which corresponds to the generator
sigma_i of the braid group.
sigma_i: a_i -> a_i a_{i+1} a_i^{-1}
a_j -> a_j, for j!=i
We assume 0<i<n, where n is the rank of ``self``.
If ``inverse`` is True returns the inverse of sigma_i.
"""
A=self._alphabet
result=dict((a,self([a])) for a in A.positive_letters())
if not inverse:
a=A[i-1]
result[a]=self([a,A[i],A.inverse_letter(a)])
result[A[i]]=self([a])
else:
a=A[i]
result[a]=self([A.inverse_letter(a),A[i-1],a])
result[A[i-1]]=self(a)
return FreeGroupAutomorphism(result,group=self)
#TODO
def random_braid(self,n=1):
"""
A random braid automorphism of ``self`` of length at most
``n``.
"""
A=self._alphabet
if n==0:
return self.identity_automorphism()
i=randint(1,len(A)-1)
j=randint(0,1)
result=self.braid_automorphism(i,j!=0)
for ii in xrange(n-1):
l=randint(0,1)
if l==j: i=randint(1,len(A)-1)
else:
k=randint(1,len(A)-2)
if j<=k: i=k+1
result=result*self.braid_automorphism(i,j)
return result
class FreeGroup_n(UniqueRepresentation, Parent):
r"""
The set of words of fixed length in a free group.
EXAMPLES::
sage: F = FreeGroup('ab')
sage: F0 = F.subset(0)
sage: F1 = F.subset(1)
sage: F2 = F.subset(2)
sage: F1.list()
[a, b, A, B]
sage: F2.list()
[aa, ab, aB, ba, bb, bA, Ab, AA, AB, Ba, BA, BB]
sage: F3.list()
[aaa,
aab,
aaB,
aba,
abb,
abA,
aBa,
...
BBB]
Standard operations with finite enumerated sets can be performed::
sage: F3.rank(F('aBa'))
6
sage: F3.unrank(6)
aBa
sage: F3.last()
BBB
"""
def __init__(self, free_group, n):
r"""
INPUT:
- ``free_group`` - a free group
- ``n`` - a non negative integer
TESTS::
sage: TestSuite(FreeGroup('ab').subset(0)).run()
sage: TestSuite(FreeGroup('abc').subset(4)).run()
"""
Parent.__init__(self, facade=free_group, category=FiniteEnumeratedSets())
self._free_group = free_group
self._n = n
def an_element(self):
r"""
Return an element in that set.
EXAMPLES::
sage: F = FreeGroup('abcd')
sage: F4 = F.subset(4)
sage: F4.an_element()
aaaa
"""
alphabet = self._free_group.alphabet()
if not alphabet:
from sage.categories.sets_cat import EmptySetError
raise EmptySetError
return self._free_group([alphabet.an_element()] * self._n, check=False)
def first(self):
r"""
Return the first element in that set.
EXAMPLES::
sage: F = FreeGroup('abcd')
sage: F4 = F.subset(4)
sage: F4.first()
aaaa
"""
alphabet = self._free_group.alphabet()
if not alphabet:
from sage.categories.sets_cat import EmptySetError
raise EmptySetError
return self._free_group([alphabet.first()] * self._n, check=False)
def last(self):
r"""
Return the last element in that set.
EXAMPLES::
sage: F = FreeGroup('abcd')
sage: F4 = F.subset(4)
sage: F4.last()
DDDD
"""
alphabet = self._free_group.alphabet()
if not alphabet:
from sage.categories.sets_cat import EmptySetError
raise EmptySetError
return self._free_group([alphabet.last()] * self._n, check=False)
def cardinality(self):
r"""
Return the cardinality of self.
EXAMPLES::
sage: F = FreeGroup('ab')
sage: F.subset(1).cardinality()
4
sage: F.subset(2).cardinality()
12
sage: F.subset(3).cardinality()
36
"""
if self._n == 0:
return 1
d = self._free_group.alphabet().cardinality()
if self._n == 1:
return d
return d * (d-1)**(self._n-1)
def random_element(self):
r"""
Return a random element in self.
EXAMPLES::
sage: F = FreeGroup(3)
sage: F.subset(3).random_element() # random
aBa
"""
if self._n == 0:
return self._free_group.one()
alphabet = self._free_group.alphabet().list()
D = len(alphabet)
d = D/2
from sage.misc.prandom import randint
j = randint(0,D-1)
data = [alphabet[j]]
while len(data) != self._n:
if j < d:
i = j + d
else:
i = j - d
j = randint(0,D-2)
if j >= i:
j += 1
data.append(alphabet[j])
return self(data, check=False)
def _repr_(self):
r"""
String representation.
TESTS::
sage: FreeGroup('abc').subset(5)
Words of length 5 in Free group over {a, b, c}
"""
return "Words of length %s in %s"%(self._n,self._free_group)
def _element_constructor_(self, data, check=True):
r"""
TESTS::
sage: F = FreeGroup('abc')
sage: G = F.subset(4)
sage: G('abaa')
abaa
sage: G('Aa')
Traceback (most recent call last):
...
ValueError: can not build a word of length 4 from given data
"""
w = self._free_group(data,check)
if check:
if len(w) != self._n:
raise ValueError("can not build a word of length %s from given data"%self._n)
return w
def __iter__(self):
r"""
Lexicographic iterator.
"""
F = self._free_group
n = self._n
if n == 0:
yield F.one()
return
alphabet = F.alphabet().list()
D = len(alphabet)
d = len(alphabet) / 2
last_pos = alphabet[d]
data = []
int_word = [] # the list of next letters
i = -1
while True:
if i == d and len(data) != n:
data.append(alphabet[1])
int_word.append(2)
while len(data) != n:
data.append(alphabet[0])
int_word.append(1)
yield F(data[:], check=False)
i = int_word.pop()
while int_word and (i == D or (i == D-1 and int_word and int_word[-1] == d)):
data.pop()
i = int_word.pop()
if int_word:
if int_word[-1] + d - 1 == i or int_word[-1] - d - 1 == i:
i += 1
elif i == D:
return
data[-1] = alphabet[i]
int_word.append(i+1)