Coverage for local/lib/python2.7/site-packages/sage/structure/parent_gens.pyx : 71%
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r""" Base class for old-style parent objects with generators
.. NOTE::
This class is being deprecated, see ``sage.structure.parent.Parent`` and ``sage.structure.category_object.CategoryObject`` for the new model.
Many parent objects in Sage are equipped with generators, which are special elements of the object. For example, the polynomial ring `\ZZ[x,y,z]` is generated by `x`, `y`, and `z`. In Sage the `i^{th}` generator of an object ``X`` is obtained using the notation ``X.gen(i)``. From the Sage interactive prompt, the shorthand notation ``X.i`` is also allowed.
REQUIRED: A class that derives from ParentWithGens *must* define the ngens() and gen(i) methods.
OPTIONAL: It is also good if they define gens() to return all gens, but this is not necessary.
The ``gens`` function returns a tuple of all generators, the ``ngens`` function returns the number of generators.
The ``_assign_names`` functions is for internal use only, and is called when objects are created to set the generator names. It can only be called once.
The following examples illustrate these functions in the context of multivariate polynomial rings and free modules.
EXAMPLES::
sage: R = PolynomialRing(ZZ, 3, 'x') sage: R.ngens() 3 sage: R.gen(0) x0 sage: R.gens() (x0, x1, x2) sage: R.variable_names() ('x0', 'x1', 'x2')
This example illustrates generators for a free module over `\ZZ`.
::
sage: M = FreeModule(ZZ, 4) sage: M Ambient free module of rank 4 over the principal ideal domain Integer Ring sage: M.ngens() 4 sage: M.gen(0) (1, 0, 0, 0) sage: M.gens() ((1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)) """
#***************************************************************************** # Copyright (C) 2005, 2006 William Stein <wstein@gmail.com> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from __future__ import absolute_import, print_function
import sage.misc.defaults from sage.misc.latex import latex_variable_name from . import gens_py cimport sage.structure.parent as parent from sage.structure.coerce_dict cimport MonoDict cimport sage.structure.category_object as category_object
cdef inline check_old_coerce(parent.Parent p):
cdef class ParentWithGens(ParentWithBase): # Derived class *must* call __init__ and set the base! def __init__(self, base, names=None, normalize=True, category = None): """ EXAMPLES::
sage: class MyParent(ParentWithGens): ....: def ngens(self): return 3 sage: P = MyParent(base = QQ, names = 'a,b,c', normalize = True, category = Groups()) sage: P.category() Category of groups sage: P._names ('a', 'b', 'c') """
# Derived class *must* define ngens method. def ngens(self): check_old_coerce(self) raise NotImplementedError("Number of generators not known.")
# Derived class *must* define gen method. def gen(self, i=0):
def gens(self): """ Return a tuple whose entries are the generators for this object, in order. """ cdef int i, n else:
def _assign_names(self, names=None, normalize=True): """ Set the names of the generator of this object.
This can only be done once because objects with generators are immutable, and is typically done during creation of the object.
EXAMPLES:
When we create this polynomial ring, self._assign_names is called by the constructor:
::
sage: R = QQ['x,y,abc']; R Multivariate Polynomial Ring in x, y, abc over Rational Field sage: R.2 abc
We can't rename the variables::
sage: R._assign_names(['a','b','c']) Traceback (most recent call last): ... ValueError: variable names cannot be changed after object creation. """ raise ValueError('variable names cannot be changed after object creation.') raise TypeError("names must be a tuple of strings")
################################################################################# # Give all objects with generators a dictionary, so that attribute setting # works. It would be nice if this functionality were standard in Pyrex, # i.e., just define __dict__ as an attribute and all this code gets generated. ################################################################################# def __getstate__(self): except AttributeError: pass
def __setstate__(self, d): except (AttributeError,KeyError): pass
################################################################################# # Morphisms of objects with generators #################################################################################
def hom(self, im_gens, codomain=None, check=True): r""" Return the unique homomorphism from self to codomain that sends ``self.gens()`` to the entries of ``im_gens``. Raises a TypeError if there is no such homomorphism.
INPUT:
- ``im_gens`` - the images in the codomain of the generators of this object under the homomorphism
- ``codomain`` - the codomain of the homomorphism
- ``check`` - whether to verify that the images of generators extend to define a map (using only canonical coercions).
OUTPUT:
- a homomorphism self --> codomain
.. NOTE::
As a shortcut, one can also give an object X instead of ``im_gens``, in which case return the (if it exists) natural map to X.
EXAMPLES: Polynomial Ring We first illustrate construction of a few homomorphisms involving a polynomial ring.
::
sage: R.<x> = PolynomialRing(ZZ) sage: f = R.hom([5], QQ) sage: f(x^2 - 19) 6
sage: R.<x> = PolynomialRing(QQ) sage: f = R.hom([5], GF(7)) Traceback (most recent call last): ... ValueError: relations do not all (canonically) map to 0 under map determined by images of generators
sage: R.<x> = PolynomialRing(GF(7)) sage: f = R.hom([3], GF(49,'a')) sage: f Ring morphism: From: Univariate Polynomial Ring in x over Finite Field of size 7 To: Finite Field in a of size 7^2 Defn: x |--> 3 sage: f(x+6) 2 sage: f(x^2+1) 3
EXAMPLES: Natural morphism
::
sage: f = ZZ.hom(GF(5)) sage: f(7) 2 sage: f Natural morphism: From: Integer Ring To: Finite Field of size 5
There might not be a natural morphism, in which case a TypeError exception is raised.
::
sage: QQ.hom(ZZ) Traceback (most recent call last): ... TypeError: natural coercion morphism from Rational Field to Integer Ring not defined """ return self.Hom(im_gens).natural_map() from sage.structure.all import Sequence im_gens = Sequence(im_gens) codomain = im_gens.universe()
cdef class localvars: r""" Context manager for safely temporarily changing the variables names of an object with generators.
Objects with named generators are globally unique in Sage. Sometimes, though, it is very useful to be able to temporarily display the generators differently. The new Python ``with`` statement and the localvars context manager make this easy and safe (and fun!)
Suppose X is any object with generators. Write
::
with localvars(X, names[, latex_names] [,normalize=False]): some code ...
and the indented code will be run as if the names in X are changed to the new names. If you give normalize=True, then the names are assumed to be a tuple of the correct number of strings.
EXAMPLES::
sage: R.<x,y> = PolynomialRing(QQ,2) sage: with localvars(R, 'z,w'): ....: print(x^3 + y^3 - x*y) z^3 + w^3 - z*w
.. NOTE::
I wrote this because it was needed to print elements of the quotient of a ring R by an ideal I using the print function for elements of R. See the code in ``quotient_ring_element.pyx``.
AUTHOR:
- William Stein (2006-10-31) """ cdef object _obj cdef object _names cdef object _latex_names cdef object _orig
def __init__(self, obj, names, latex_names=None, normalize=True): else:
def __enter__(self):
def __exit__(self, type, value, traceback):
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