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free_abelian_monoid_element.py
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free_abelian_monoid_element.py
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"""
Abelian Monoid Elements
AUTHORS:
- David Kohel (2005-09)
EXAMPLES:
Recall the example from abelian monoids::
sage: F = FreeAbelianMonoid(5,names = list("abcde"))
sage: (a,b,c,d,e) = F.gens()
sage: a*b^2*e*d
a*b^2*d*e
sage: x = b^2*e*d*a^7
sage: x
a^7*b^2*d*e
sage: x.list()
[7, 2, 0, 1, 1]
The list is a copy, so changing the list does not change the element::
sage: x.list()[0] = 0
sage: x
a^7*b^2*d*e
"""
#*****************************************************************************
# Copyright (C) 2006 William Stein <wstein@gmail.com>
# Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
from six import integer_types
from sage.structure.sage_object import richcmp
from sage.rings.integer import Integer
from sage.structure.element import MonoidElement
def is_FreeAbelianMonoidElement(x):
r"""
Queries whether ``x`` is an object of type ``FreeAbelianMonoidElement``.
INPUT:
- ``x`` -- an object.
OUTPUT:
- ``True`` if ``x`` is an object of type ``FreeAbelianMonoidElement``;
``False`` otherwise.
"""
return isinstance(x, FreeAbelianMonoidElement)
class FreeAbelianMonoidElement(MonoidElement):
def __init__(self, F, x):
"""
Create the element x of the FreeAbelianMonoid F.
EXAMPLES::
sage: F = FreeAbelianMonoid(5, 'abcde')
sage: F
Free abelian monoid on 5 generators (a, b, c, d, e)
sage: F(1)
1
sage: a, b, c, d, e = F.gens()
sage: a^2 * b^3 * a^2 * b^4
a^4*b^7
sage: F = FreeAbelianMonoid(5, 'abcde')
sage: a, b, c, d, e = F.gens()
sage: a in F
True
sage: a*b in F
True
"""
MonoidElement.__init__(self, F)
n = F.ngens()
if isinstance(x, integer_types + (Integer,)) and x == 1:
self._element_vector = tuple([0]*n)
elif isinstance(x, (list, tuple)):
if len(x) != n:
raise IndexError("argument length (= %s) must be %s"%(len(x), n))
self._element_vector = tuple(x)
else:
raise TypeError("argument x (= %s) is of wrong type"%x)
def _repr_(self):
"""
Return a string representation of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(5, 'abcde')
sage: F(1)
1
sage: a, b, c, d, e = F.gens()
sage: a^2 * b^3 * a^2 * b^4
a^4*b^7
"""
s = ""
A = self.parent()
n = A.ngens()
x = A.variable_names()
v = self._element_vector
for i in range(n):
if v[i] == 0:
continue
elif v[i] == 1:
if len(s) > 0: s += "*"
s += "%s"%x[i]
else:
if len(s) > 0: s += "*"
s += "%s^%s"%(x[i],v[i])
if not s:
s = "1"
return s
def _richcmp_(self, other, op):
"""
Rich comparison.
EXAMPLES::
sage: F = FreeAbelianMonoid(5, 'abcde')
sage: F(1)
1
sage: a, b, c, d, e = F.gens()
sage: x = a^2 * b^3
sage: F(1) < x
True
sage: x > b
True
sage: x <= a^4
True
sage: x != a*b
True
sage: a*b == b*a
True
sage: x > a^3*b^2
False
"""
return richcmp(self._element_vector, other._element_vector, op)
def __mul__(self, y):
if not isinstance(y, FreeAbelianMonoidElement):
raise TypeError("argument y (= %s) is of wrong type"%y)
M = self.parent()
z = M.element_class(M, 1)
xelt = self._element_vector
yelt = y._element_vector
z._element_vector = tuple([xelt[i]+yelt[i] for i in range(len(xelt))])
return z
def __pow__(self, n):
"""
Raises self to the power of `n`.
AUTHORS:
- Tom Boothby (2007-08): Replaced O(log n) time, O(n) space
algorithm with O(1) time and space"algorithm".
EXAMPLES::
sage: F = FreeAbelianMonoid(5,names = list("abcde"))
sage: (a,b,c,d,e) = F.gens()
sage: x = a*b^2*e*d; x
a*b^2*d*e
sage: x^3
a^3*b^6*d^3*e^3
sage: x^0
1
"""
if not isinstance(n, integer_types + (Integer,)):
raise TypeError("argument n (= %s) must be an integer"%(n,))
if n < 0:
raise IndexError("argument n (= %s) must be positive"%n)
elif n == 1:
return self
M = self.parent()
z = M.element_class(M, 1)
if n == 0:
return z
else:
z._element_vector = tuple([i*n for i in self._element_vector])
return z
def __hash__(self):
"""
Return the hash of ``self``.
EXAMPLES::
sage: F = FreeAbelianMonoid(5,names = list("abcde"))
sage: (a,b,c,d,e) = F.gens()
sage: x = a*b^2*e*d
sage: hash(x) == hash(x)
True
"""
return hash(self._element_vector)
def list(self):
"""
Return (a reference to) the underlying list used to represent this
element. If this is a monoid in an abelian monoid on `n`
generators, then this is a list of nonnegative integers of length
`n`.
EXAMPLES::
sage: F = FreeAbelianMonoid(5, 'abcde')
sage: (a, b, c, d, e) = F.gens()
sage: a.list()
[1, 0, 0, 0, 0]
"""
return list(self._element_vector)