Module: sage.structure.parent_gens
Base class for parent objects with generators.
Many parent objects in SAGE are equipped with generators, which are
special elements of the object. For example, the polynomial ring
is generated by
,
, and
. In SAGE the
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.
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
.
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))
Module-level Functions
) |
Return True if x is a parent object with additive abelian generators, i.e., derives from sage.structure.parent.ParentWithAdditiveAbelianGens and False otherwise.
sage: is_ParentWithAdditiveAbelianGens(QQ) False sage: is_ParentWithAdditiveAbelianGens(QQ^3) True
) |
Return True if x is a parent object with generators, i.e., derives from sage.structure.parent.ParentWithGens and False otherwise.
sage: is_ParentWithGens(QQ['x']) True sage: is_ParentWithGens(CC) True sage: is_ParentWithGens(Primes()) False
) |
Return True if x is a parent object with additive abelian generators, i.e., derives from sage.structure.parent.ParentWithMultiplicativeAbelianGens and False otherwise.
sage: is_ParentWithMultiplicativeAbelianGens(QQ) False sage: is_ParentWithMultiplicativeAbelianGens(DirichletGroup(11)) True
) |
Class: localvars
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 ...
If you're writing Python library code, you currently have
to put from __future__ import with_statement
in your file
in order to use the with
statement. This restriction will
disappear in Python 2.6.
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
NOTES: 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)
Special Functions: __enter__,
__exit__,
__init__
) |
) |
) |
Class: ParentWithAdditiveAbelianGens
Functions: generator_orders
Special Functions: __iter__
) |
Return an iterator over the elements in this object.
Class: ParentWithGens
Functions: gen,
gens,
gens_dict,
hom,
inject_variables,
injvar,
latex_name,
latex_variable_names,
list,
ngens,
objgen,
objgens,
variable_name,
variable_names
) |
Return a tuple whose entries are the generators for this object, in order.
) |
Return a dictionary whose entries are var_name:variable,...
.
) |
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:
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.
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): ... TypeError: images do not define a valid homomorphism
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
Natural morphism
sage: f = ZZ.hom(GF(5)) sage: f(7) 2 sage: f Ring Coercion 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.
) |
Inject the generators of self with their names into the namespace of the Python code from which this function is called. Thus, e.g., if the generators of self are labeled 'a', 'b', and 'c', then after calling this method the variables a, b, and c in the current scope will be set equal to the generators of self.
NOTE: If Foo is a constructor for a SAGE object with generators, and Foo is defined in Pyrex, then it would typically call inject_variables() on the object it creates. E.g., PolyomialRing(QQ, 'y') does this so that the variable y is the generator of the polynomial ring.
) |
This is a synonym for self.inject_variables(...) «<sage.structure.parent_gens.ParentWithGens.inject_variables»>
) |
Returns the list of variable names suitable for latex output.
All '_SOMETHING' substrings are replaced by '_SOMETHING' recursively so that subscripts of subscripts work.
sage: R, x = PolynomialRing(QQ,'x',12).objgens() sage: x (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) sage: print R.latex_variable_names () ['x_{0}', 'x_{1}', 'x_{2}', 'x_{3}', 'x_{4}', 'x_{5}', 'x_{6}', 'x_{7}', 'x_{8}', 'x_{9}', 'x_{10}', 'x_{11}'] sage: f = x[0]^3 + 15/3 * x[1]^10 sage: print latex(f) 5 x_{1}^{10} + x_{0}^{3}
) |
Return a list of all elements in this object, if possible (the object must define an iterator).
) |
Return self and the generator of self.
Input:
sage: R, x = PolynomialRing(QQ,'x').objgen() sage: R Univariate Polynomial Ring in x over Rational Field sage: x x
) |
Return self and the generators of self as a tuple.
Input:
sage: R, vars = PolynomialRing(QQ,3, 'x').objgens() sage: R Multivariate Polynomial Ring in x0, x1, x2 over Rational Field sage: vars (x0, x1, x2)
Special Functions: __getitem__,
__getslice__,
__getstate__,
__init__,
__len__,
__setstate__,
__temporarily_change_names,
_assign_names,
_first_ngens,
_is_valid_homomorphism_
) |
) |
) |
) |
This is used by the variable names context manager.
) |
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.
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.
) |
) |
Return True if im_gens
defines a valid homomorphism
from self to codomain; otherwise return False.
If determining whether or not a homomorphism is valid has not been implemented for this ring, then a NotImplementedError exception is raised.
Class: ParentWithMultiplicativeAbelianGens
Functions: generator_orders
Special Functions: __iter__
) |
Return an iterator over the elements in this object.
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