Base class for oldstyle 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))

class
sage.structure.parent_gens.
ParentWithGens
¶ Bases:
sage.structure.parent_base.ParentWithBase
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')

gen
(i=0)¶

gens
()¶ Return a tuple whose entries are the generators for this object, in order.

hom
(im_gens, codomain=None, check=True)¶ Return the unique homomorphism from self to codomain that sends
self.gens()
to the entries ofim_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 homomorphismcodomain
 the codomain of the homomorphismcheck
 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): ... 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
EXAMPLES: 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.

ngens
()¶


class
sage.structure.parent_gens.
localvars
¶ Bases:
object
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 (20061031)