Coercion via construction functors#

class sage.categories.pushout.AlgebraicClosureFunctor[source]#

Bases: ConstructionFunctor

Algebraic Closure.

EXAMPLES:

sage: # needs sage.rings.complex_double sage.rings.number_field
sage: F = CDF.construction()[0]
sage: F(QQ)
Algebraic Field
sage: F(RR)                                                                     # needs sage.rings.real_mpfr
Complex Field with 53 bits of precision
sage: F(F(QQ)) is F(QQ)
True

>>> from sage.all import *
>>> # needs sage.rings.complex_double sage.rings.number_field
>>> F = CDF.construction()[Integer(0)]
>>> F(QQ)
Algebraic Field
>>> F(RR)                                                                     # needs sage.rings.real_mpfr
Complex Field with 53 bits of precision
>>> F(F(QQ)) is F(QQ)
True
merge(other)[source]#

Mathematically, Algebraic Closure subsumes Algebraic Extension. However, it seems that people do want to work with algebraic extensions of RR. Therefore, we do not merge with algebraic extension.

rank = 3#
class sage.categories.pushout.AlgebraicExtensionFunctor(polys, names, embeddings=None, structures=None, cyclotomic=None, precs=None, implementations=None, *, residue=None, latex_names=None, **kwds)[source]#

Bases: ConstructionFunctor

Algebraic extension (univariate polynomial ring modulo principal ideal).

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 + x^2 + 1)                                        # needs sage.rings.number_field
sage: F = K.construction()[0]                                                   # needs sage.rings.number_field
sage: F(ZZ['t'])                                                                # needs sage.rings.number_field
Univariate Quotient Polynomial Ring in a
 over Univariate Polynomial Ring in t over Integer Ring
 with modulus a^3 + a^2 + 1

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> F = K.construction()[Integer(0)]                                                   # needs sage.rings.number_field
>>> F(ZZ['t'])                                                                # needs sage.rings.number_field
Univariate Quotient Polynomial Ring in a
 over Univariate Polynomial Ring in t over Integer Ring
 with modulus a^3 + a^2 + 1

Note that, even if a field is algebraically closed, the algebraic extension will be constructed as the quotient of a univariate polynomial ring:

sage: F(CC)                                                                     # needs sage.rings.number_field
Univariate Quotient Polynomial Ring in a
 over Complex Field with 53 bits of precision
 with modulus a^3 + a^2 + 1.00000000000000
sage: F(RR)                                                                     # needs sage.rings.number_field
Univariate Quotient Polynomial Ring in a
 over Real Field with 53 bits of precision
 with modulus a^3 + a^2 + 1.00000000000000

>>> from sage.all import *
>>> F(CC)                                                                     # needs sage.rings.number_field
Univariate Quotient Polynomial Ring in a
 over Complex Field with 53 bits of precision
 with modulus a^3 + a^2 + 1.00000000000000
>>> F(RR)                                                                     # needs sage.rings.number_field
Univariate Quotient Polynomial Ring in a
 over Real Field with 53 bits of precision
 with modulus a^3 + a^2 + 1.00000000000000

Note that the construction functor of a number field applied to the integers returns an order (not necessarily maximal) of that field, similar to the behaviour of ZZ.extension(...):

sage: F(ZZ)                                                                     # needs sage.rings.number_field
Order generated by a in Number Field in a with defining polynomial x^3 + x^2 + 1

>>> from sage.all import *
>>> F(ZZ)                                                                     # needs sage.rings.number_field
Order generated by a in Number Field in a with defining polynomial x^3 + x^2 + 1

This also holds for non-absolute number fields:

sage: # needs sage.rings.number_field
sage: x = polygen(QQ, 'x')
sage: K.<a,b> = NumberField([x^3 + x^2 + 1, x^2 + x + 1])
sage: F = K.construction()[0]
sage: O = F(ZZ); O
Relative Order
 generated by [(b - 2)*a^2 + (3*b - 1)*a + 3*b + 4, a - b]
 in Number Field in a with defining polynomial x^3 + x^2 + 1
 over its base field
sage: O.ambient() is K
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(QQ, 'x')
>>> K = NumberField([x**Integer(3) + x**Integer(2) + Integer(1), x**Integer(2) + x + Integer(1)], names=('a', 'b',)); (a, b,) = K._first_ngens(2)
>>> F = K.construction()[Integer(0)]
>>> O = F(ZZ); O
Relative Order
 generated by [(b - 2)*a^2 + (3*b - 1)*a + 3*b + 4, a - b]
 in Number Field in a with defining polynomial x^3 + x^2 + 1
 over its base field
>>> O.ambient() is K
True

Special cases are made for cyclotomic fields and residue fields:

sage: # needs sage.rings.number_field
sage: C = CyclotomicField(8)
sage: F, R = C.construction()
sage: F
AlgebraicExtensionFunctor
sage: R
Rational Field
sage: F(R)
Cyclotomic Field of order 8 and degree 4
sage: F(ZZ)
Maximal Order generated by zeta8 in Cyclotomic Field of order 8 and degree 4

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> C = CyclotomicField(Integer(8))
>>> F, R = C.construction()
>>> F
AlgebraicExtensionFunctor
>>> R
Rational Field
>>> F(R)
Cyclotomic Field of order 8 and degree 4
>>> F(ZZ)
Maximal Order generated by zeta8 in Cyclotomic Field of order 8 and degree 4

sage: # needs sage.rings.number_field
sage: K.<z> = CyclotomicField(7)
sage: P = K.factor(17)[0][0]
sage: k = K.residue_field(P)
sage: F, R = k.construction()
sage: F
AlgebraicExtensionFunctor
sage: R
Cyclotomic Field of order 7 and degree 6
sage: F(R) is k
True
sage: F(ZZ)
Residue field of Integers modulo 17
sage: F(CyclotomicField(49))
Residue field in zbar of Fractional ideal (17)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(7), names=('z',)); (z,) = K._first_ngens(1)
>>> P = K.factor(Integer(17))[Integer(0)][Integer(0)]
>>> k = K.residue_field(P)
>>> F, R = k.construction()
>>> F
AlgebraicExtensionFunctor
>>> R
Cyclotomic Field of order 7 and degree 6
>>> F(R) is k
True
>>> F(ZZ)
Residue field of Integers modulo 17
>>> F(CyclotomicField(Integer(49)))
Residue field in zbar of Fractional ideal (17)
expand()[source]#

Decompose the functor \(F\) into sub-functors, whose product returns \(F\).

EXAMPLES:

sage: # needs sage.rings.number_field
sage: P.<x> = QQ[]
sage: K.<a> = NumberField(x^3 - 5, embedding=0)
sage: L.<b> = K.extension(x^2 + a)
sage: F, R = L.construction()
sage: prod(F.expand())(R) == L
True
sage: K = NumberField([x^2 - 2, x^2 - 3],'a')
sage: F, R = K.construction()
sage: F
AlgebraicExtensionFunctor
sage: L = F.expand(); L
[AlgebraicExtensionFunctor, AlgebraicExtensionFunctor]
sage: L[-1](QQ)
Number Field in a1 with defining polynomial x^2 - 3

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> P = QQ['x']; (x,) = P._first_ngens(1)
>>> K = NumberField(x**Integer(3) - Integer(5), embedding=Integer(0), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.extension(x**Integer(2) + a, names=('b',)); (b,) = L._first_ngens(1)
>>> F, R = L.construction()
>>> prod(F.expand())(R) == L
True
>>> K = NumberField([x**Integer(2) - Integer(2), x**Integer(2) - Integer(3)],'a')
>>> F, R = K.construction()
>>> F
AlgebraicExtensionFunctor
>>> L = F.expand(); L
[AlgebraicExtensionFunctor, AlgebraicExtensionFunctor]
>>> L[-Integer(1)](QQ)
Number Field in a1 with defining polynomial x^2 - 3
merge(other)[source]#

Merging with another AlgebraicExtensionFunctor.

INPUT:

  • other – Construction Functor.

OUTPUT:

  • If self==other, self is returned.

  • If self and other are simple extensions and both provide an embedding, then it is tested whether one of the number fields provided by the functors coerces into the other; the functor associated with the target of the coercion is returned. Otherwise, the construction functor associated with the pushout of the codomains of the two embeddings is returned, provided that it is a number field.

  • If these two extensions are defined by Conway polynomials over finite fields, merges them into a single extension of degree the lcm of the two degrees.

  • Otherwise, None is returned.

REMARK:

Algebraic extension with embeddings currently only works when applied to the rational field. This is why we use the admittedly strange rule above for merging.

EXAMPLES:

The following demonstrate coercions for finite fields using Conway or pseudo-Conway polynomials:

sage: k = GF(3^2, prefix='z'); a = k.gen()                                  # needs sage.rings.finite_rings
sage: l = GF(3^3, prefix='z'); b = l.gen()                                  # needs sage.rings.finite_rings
sage: a + b  # indirect doctest                                             # needs sage.rings.finite_rings
z6^5 + 2*z6^4 + 2*z6^3 + z6^2 + 2*z6 + 1

>>> from sage.all import *
>>> k = GF(Integer(3)**Integer(2), prefix='z'); a = k.gen()                                  # needs sage.rings.finite_rings
>>> l = GF(Integer(3)**Integer(3), prefix='z'); b = l.gen()                                  # needs sage.rings.finite_rings
>>> a + b  # indirect doctest                                             # needs sage.rings.finite_rings
z6^5 + 2*z6^4 + 2*z6^3 + z6^2 + 2*z6 + 1

Note that embeddings are compatible in lattices of such finite fields:

sage: # needs sage.rings.finite_rings
sage: m = GF(3^5, prefix='z'); c = m.gen()
sage: (a + b) + c == a + (b + c) # indirect doctest
True
sage: from sage.categories.pushout import pushout
sage: n = pushout(k, l)
sage: o = pushout(l, m)
sage: q = pushout(n, o)
sage: q(o(b)) == q(n(b)) # indirect doctest
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> m = GF(Integer(3)**Integer(5), prefix='z'); c = m.gen()
>>> (a + b) + c == a + (b + c) # indirect doctest
True
>>> from sage.categories.pushout import pushout
>>> n = pushout(k, l)
>>> o = pushout(l, m)
>>> q = pushout(n, o)
>>> q(o(b)) == q(n(b)) # indirect doctest
True

Coercion is also available for number fields:

sage: # needs sage.rings.number_field
sage: P.<x> = QQ[]
sage: L.<b> = NumberField(x^8 - x^4 + 1, embedding=CDF.0)
sage: M1.<c1> = NumberField(x^2 + x + 1, embedding=b^4 - 1)
sage: M2.<c2> = NumberField(x^2 + 1, embedding=-b^6)
sage: M1.coerce_map_from(M2)
sage: M2.coerce_map_from(M1)
sage: c1 + c2; parent(c1 + c2)    #indirect doctest
-b^6 + b^4 - 1
Number Field in b with defining polynomial x^8 - x^4 + 1
 with b = -0.2588190451025208? + 0.9659258262890683?*I
sage: pushout(M1['x'], M2['x'])                                             # needs sage.rings.finite_rings
Univariate Polynomial Ring in x
 over Number Field in b with defining polynomial x^8 - x^4 + 1
  with b = -0.2588190451025208? + 0.9659258262890683?*I

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> P = QQ['x']; (x,) = P._first_ngens(1)
>>> L = NumberField(x**Integer(8) - x**Integer(4) + Integer(1), embedding=CDF.gen(0), names=('b',)); (b,) = L._first_ngens(1)
>>> M1 = NumberField(x**Integer(2) + x + Integer(1), embedding=b**Integer(4) - Integer(1), names=('c1',)); (c1,) = M1._first_ngens(1)
>>> M2 = NumberField(x**Integer(2) + Integer(1), embedding=-b**Integer(6), names=('c2',)); (c2,) = M2._first_ngens(1)
>>> M1.coerce_map_from(M2)
>>> M2.coerce_map_from(M1)
>>> c1 + c2; parent(c1 + c2)    #indirect doctest
-b^6 + b^4 - 1
Number Field in b with defining polynomial x^8 - x^4 + 1
 with b = -0.2588190451025208? + 0.9659258262890683?*I
>>> pushout(M1['x'], M2['x'])                                             # needs sage.rings.finite_rings
Univariate Polynomial Ring in x
 over Number Field in b with defining polynomial x^8 - x^4 + 1
  with b = -0.2588190451025208? + 0.9659258262890683?*I

In the previous example, the number field L becomes the pushout of M1 and M2 since both are provided with an embedding into L, and since L is a number field. If two number fields are embedded into a field that is not a numberfield, no merging occurs:

sage: # needs sage.rings.complex_double sage.rings.number_field
sage: cbrt2 = CDF(2)^(1/3)
sage: zeta3 = CDF.zeta(3)
sage: K.<a> = NumberField(x^3 - 2, embedding=cbrt2 * zeta3)
sage: L.<b> = NumberField(x^6 - 2, embedding=1.1)
sage: L.coerce_map_from(K)
sage: K.coerce_map_from(L)
sage: pushout(K, L)                                                         # needs sage.rings.finite_rings
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension', Number Field in a with
defining polynomial x^3 - 2 with a = -0.6299605249474365? + 1.091123635971722?*I,
Number Field in b with defining polynomial x^6 - 2 with b = 1.122462048309373?)

>>> from sage.all import *
>>> # needs sage.rings.complex_double sage.rings.number_field
>>> cbrt2 = CDF(Integer(2))**(Integer(1)/Integer(3))
>>> zeta3 = CDF.zeta(Integer(3))
>>> K = NumberField(x**Integer(3) - Integer(2), embedding=cbrt2 * zeta3, names=('a',)); (a,) = K._first_ngens(1)
>>> L = NumberField(x**Integer(6) - Integer(2), embedding=RealNumber('1.1'), names=('b',)); (b,) = L._first_ngens(1)
>>> L.coerce_map_from(K)
>>> K.coerce_map_from(L)
>>> pushout(K, L)                                                         # needs sage.rings.finite_rings
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension', Number Field in a with
defining polynomial x^3 - 2 with a = -0.6299605249474365? + 1.091123635971722?*I,
Number Field in b with defining polynomial x^6 - 2 with b = 1.122462048309373?)
rank = 3#
class sage.categories.pushout.BlackBoxConstructionFunctor(box)[source]#

Bases: ConstructionFunctor

Construction functor obtained from any callable object.

EXAMPLES:

sage: from sage.categories.pushout import BlackBoxConstructionFunctor

sage: # needs sage.libs.gap
sage: from sage.interfaces.gap import gap
sage: FG = BlackBoxConstructionFunctor(gap)
sage: FG
BlackBoxConstructionFunctor
sage: FG(ZZ)
Integers
sage: FG(ZZ).parent()
Gap
sage: FG == loads(dumps(FG))
True

sage: FS = BlackBoxConstructionFunctor(singular)
sage: FS(QQ['t'])                                                               # needs sage.libs.singular
polynomial ring, over a field, global ordering
//   coefficients: QQ
//   number of vars : 1
//        block   1 : ordering lp
//                  : names    t
//        block   2 : ordering C
sage: FG == FS                                                                  # needs sage.libs.gap sage.libs.singular
False

>>> from sage.all import *
>>> from sage.categories.pushout import BlackBoxConstructionFunctor

>>> # needs sage.libs.gap
>>> from sage.interfaces.gap import gap
>>> FG = BlackBoxConstructionFunctor(gap)
>>> FG
BlackBoxConstructionFunctor
>>> FG(ZZ)
Integers
>>> FG(ZZ).parent()
Gap
>>> FG == loads(dumps(FG))
True

>>> FS = BlackBoxConstructionFunctor(singular)
>>> FS(QQ['t'])                                                               # needs sage.libs.singular
polynomial ring, over a field, global ordering
//   coefficients: QQ
//   number of vars : 1
//        block   1 : ordering lp
//                  : names    t
//        block   2 : ordering C
>>> FG == FS                                                                  # needs sage.libs.gap sage.libs.singular
False
rank = 100#
class sage.categories.pushout.CompletionFunctor(p, prec, extras=None)[source]#

Bases: ConstructionFunctor

Completion of a ring with respect to a given prime (including infinity).

EXAMPLES:

sage: # needs sage.rings.padics
sage: R = Zp(5)
sage: R
5-adic Ring with capped relative precision 20
sage: F1 = R.construction()[0]
sage: F1
Completion[5, prec=20]
sage: F1(ZZ) is R
True
sage: F1(QQ)
5-adic Field with capped relative precision 20

sage: F2 = RR.construction()[0]
sage: F2
Completion[+Infinity, prec=53]
sage: F2(QQ) is RR
True

sage: P.<x> = ZZ[]
sage: Px = P.completion(x) # currently the only implemented completion of P
sage: Px
Power Series Ring in x over Integer Ring
sage: F3 = Px.construction()[0]
sage: F3(GF(3)['x'])
Power Series Ring in x over Finite Field of size 3

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> R = Zp(Integer(5))
>>> R
5-adic Ring with capped relative precision 20
>>> F1 = R.construction()[Integer(0)]
>>> F1
Completion[5, prec=20]
>>> F1(ZZ) is R
True
>>> F1(QQ)
5-adic Field with capped relative precision 20

>>> F2 = RR.construction()[Integer(0)]
>>> F2
Completion[+Infinity, prec=53]
>>> F2(QQ) is RR
True

>>> P = ZZ['x']; (x,) = P._first_ngens(1)
>>> Px = P.completion(x) # currently the only implemented completion of P
>>> Px
Power Series Ring in x over Integer Ring
>>> F3 = Px.construction()[Integer(0)]
>>> F3(GF(Integer(3))['x'])
Power Series Ring in x over Finite Field of size 3
commutes(other)[source]#

Completion commutes with fraction fields.

EXAMPLES:

sage: F1 = Zp(5).construction()[0]                                          # needs sage.rings.padics
sage: F2 = QQ.construction()[0]
sage: F1.commutes(F2)                                                       # needs sage.rings.padics
True

>>> from sage.all import *
>>> F1 = Zp(Integer(5)).construction()[Integer(0)]                                          # needs sage.rings.padics
>>> F2 = QQ.construction()[Integer(0)]
>>> F1.commutes(F2)                                                       # needs sage.rings.padics
True
merge(other)[source]#

Two Completion functors are merged, if they are equal. If the precisions of both functors coincide, then a Completion functor is returned that results from updating the extras dictionary of self by other.extras. Otherwise, if the completion is at infinity then merging does not increase the set precision, and if the completion is at a finite prime, merging does not decrease the capped precision.

EXAMPLES:

sage: # needs sage.rings.padics
sage: R1.<a> = Zp(5, prec=20)[]
sage: R2 = Qp(5, prec=40)
sage: R2(1) + a       # indirect doctest
(1 + O(5^20))*a + 1 + O(5^40)
sage: R3 = RealField(30)
sage: R4 = RealField(50)
sage: R3(1) + R4(1)   # indirect doctest
2.0000000
sage: (R3(1) + R4(1)).parent()
Real Field with 30 bits of precision

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> R1 = Zp(Integer(5), prec=Integer(20))['a']; (a,) = R1._first_ngens(1)
>>> R2 = Qp(Integer(5), prec=Integer(40))
>>> R2(Integer(1)) + a       # indirect doctest
(1 + O(5^20))*a + 1 + O(5^40)
>>> R3 = RealField(Integer(30))
>>> R4 = RealField(Integer(50))
>>> R3(Integer(1)) + R4(Integer(1))   # indirect doctest
2.0000000
>>> (R3(Integer(1)) + R4(Integer(1))).parent()
Real Field with 30 bits of precision
rank = 4#
class sage.categories.pushout.CompositeConstructionFunctor(*args)[source]#

Bases: ConstructionFunctor

A Construction Functor composed by other Construction Functors.

INPUT:

F1, F2,...: A list of Construction Functors. The result is the composition F1 followed by F2 followed by …

EXAMPLES:

sage: from sage.categories.pushout import CompositeConstructionFunctor
sage: F = CompositeConstructionFunctor(QQ.construction()[0], ZZ['x'].construction()[0],
....:                                  QQ.construction()[0], ZZ['y'].construction()[0])
sage: F
Poly[y](FractionField(Poly[x](FractionField(...))))
sage: F == loads(dumps(F))
True
sage: F == CompositeConstructionFunctor(*F.all)
True
sage: F(GF(2)['t'])                                                             # needs sage.libs.ntl
Univariate Polynomial Ring in y
 over Fraction Field of Univariate Polynomial Ring in x
  over Fraction Field of Univariate Polynomial Ring in t
   over Finite Field of size 2 (using GF2X)

>>> from sage.all import *
>>> from sage.categories.pushout import CompositeConstructionFunctor
>>> F = CompositeConstructionFunctor(QQ.construction()[Integer(0)], ZZ['x'].construction()[Integer(0)],
...                                  QQ.construction()[Integer(0)], ZZ['y'].construction()[Integer(0)])
>>> F
Poly[y](FractionField(Poly[x](FractionField(...))))
>>> F == loads(dumps(F))
True
>>> F == CompositeConstructionFunctor(*F.all)
True
>>> F(GF(Integer(2))['t'])                                                             # needs sage.libs.ntl
Univariate Polynomial Ring in y
 over Fraction Field of Univariate Polynomial Ring in x
  over Fraction Field of Univariate Polynomial Ring in t
   over Finite Field of size 2 (using GF2X)
expand()[source]#

Return expansion of a CompositeConstructionFunctor.

Note

The product over the list of components, as returned by the expand() method, is equal to self.

EXAMPLES:

sage: from sage.categories.pushout import CompositeConstructionFunctor
sage: F = CompositeConstructionFunctor(QQ.construction()[0],
....:                                  ZZ['x'].construction()[0],
....:                                  QQ.construction()[0],
....:                                  ZZ['y'].construction()[0])
sage: F
Poly[y](FractionField(Poly[x](FractionField(...))))
sage: prod(F.expand()) == F
True

>>> from sage.all import *
>>> from sage.categories.pushout import CompositeConstructionFunctor
>>> F = CompositeConstructionFunctor(QQ.construction()[Integer(0)],
...                                  ZZ['x'].construction()[Integer(0)],
...                                  QQ.construction()[Integer(0)],
...                                  ZZ['y'].construction()[Integer(0)])
>>> F
Poly[y](FractionField(Poly[x](FractionField(...))))
>>> prod(F.expand()) == F
True
class sage.categories.pushout.ConstructionFunctor[source]#

Bases: Functor

Base class for construction functors.

A construction functor is a functorial algebraic construction, such as the construction of a matrix ring over a given ring or the fraction field of a given ring.

In addition to the class Functor, construction functors provide rules for combining and merging constructions. This is an important part of Sage’s coercion model, namely the pushout of two constructions: When a polynomial p in a variable x with integer coefficients is added to a rational number q, then Sage finds that the parents ZZ['x'] and QQ are obtained from ZZ by applying a polynomial ring construction respectively the fraction field construction. Each construction functor has an attribute rank, and the rank of the polynomial ring construction is higher than the rank of the fraction field construction. This means that the pushout of QQ and ZZ['x'], and thus a common parent in which p and q can be added, is QQ['x'], since the construction functor with a lower rank is applied first.

sage: F1, R = QQ.construction()
sage: F1
FractionField
sage: R
Integer Ring
sage: F2, R = (ZZ['x']).construction()
sage: F2
Poly[x]
sage: R
Integer Ring
sage: F3 = F2.pushout(F1)
sage: F3
Poly[x](FractionField(...))
sage: F3(R)
Univariate Polynomial Ring in x over Rational Field
sage: from sage.categories.pushout import pushout
sage: P.<x> = ZZ[]
sage: pushout(QQ,P)
Univariate Polynomial Ring in x over Rational Field
sage: ((x+1) + 1/2).parent()
Univariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> F1, R = QQ.construction()
>>> F1
FractionField
>>> R
Integer Ring
>>> F2, R = (ZZ['x']).construction()
>>> F2
Poly[x]
>>> R
Integer Ring
>>> F3 = F2.pushout(F1)
>>> F3
Poly[x](FractionField(...))
>>> F3(R)
Univariate Polynomial Ring in x over Rational Field
>>> from sage.categories.pushout import pushout
>>> P = ZZ['x']; (x,) = P._first_ngens(1)
>>> pushout(QQ,P)
Univariate Polynomial Ring in x over Rational Field
>>> ((x+Integer(1)) + Integer(1)/Integer(2)).parent()
Univariate Polynomial Ring in x over Rational Field

When composing two construction functors, they are sometimes merged into one, as is the case in the Quotient construction:

sage: Q15, R = (ZZ.quo(15*ZZ)).construction()
sage: Q15
QuotientFunctor
sage: Q35, R = (ZZ.quo(35*ZZ)).construction()
sage: Q35
QuotientFunctor
sage: Q15.merge(Q35)
QuotientFunctor
sage: Q15.merge(Q35)(ZZ)
Ring of integers modulo 5

>>> from sage.all import *
>>> Q15, R = (ZZ.quo(Integer(15)*ZZ)).construction()
>>> Q15
QuotientFunctor
>>> Q35, R = (ZZ.quo(Integer(35)*ZZ)).construction()
>>> Q35
QuotientFunctor
>>> Q15.merge(Q35)
QuotientFunctor
>>> Q15.merge(Q35)(ZZ)
Ring of integers modulo 5

Functors can not only be applied to objects, but also to morphisms in the respective categories. For example:

sage: P.<x,y> = ZZ[]
sage: F = P.construction()[0]; F
MPoly[x,y]
sage: A.<a,b> = GF(5)[]
sage: f = A.hom([a + b, a - b], A)
sage: F(A)
Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
sage: F(f)
Ring endomorphism of Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
  Defn: Induced from base ring by
        Ring endomorphism of Multivariate Polynomial Ring in a, b
         over Finite Field of size 5
          Defn: a |--> a + b
                b |--> a - b
sage: F(f)(F(A)(x)*a)
(a + b)*x

>>> from sage.all import *
>>> P = ZZ['x, y']; (x, y,) = P._first_ngens(2)
>>> F = P.construction()[Integer(0)]; F
MPoly[x,y]
>>> A = GF(Integer(5))['a, b']; (a, b,) = A._first_ngens(2)
>>> f = A.hom([a + b, a - b], A)
>>> F(A)
Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
>>> F(f)
Ring endomorphism of Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
  Defn: Induced from base ring by
        Ring endomorphism of Multivariate Polynomial Ring in a, b
         over Finite Field of size 5
          Defn: a |--> a + b
                b |--> a - b
>>> F(f)(F(A)(x)*a)
(a + b)*x
coercion_reversed = False#
common_base(other_functor, self_bases, other_bases)[source]#

This function is called by pushout() when no common parent is found in the construction tower.

Note

The main use is for multivariate construction functors, which use this function to implement recursion for pushout().

INPUT:

  • other_functor – a construction functor.

  • self_bases – the arguments passed to this functor.

  • other_bases – the arguments passed to the functor other_functor.

OUTPUT:

Nothing, since a CoercionException is raised.

Note

Overload this function in derived class, see e.e. MultivariateConstructionFunctor.

commutes(other)[source]#

Determine whether self commutes with another construction functor.

Note

By default, False is returned in all cases (even if the two functors are the same, since in this case merge() will apply anyway). So far there is no construction functor that overloads this method. Anyway, this method only becomes relevant if two construction functors have the same rank.

EXAMPLES:

sage: F = QQ.construction()[0]
sage: P = ZZ['t'].construction()[0]
sage: F.commutes(P)
False
sage: P.commutes(F)
False
sage: F.commutes(F)
False

>>> from sage.all import *
>>> F = QQ.construction()[Integer(0)]
>>> P = ZZ['t'].construction()[Integer(0)]
>>> F.commutes(P)
False
>>> P.commutes(F)
False
>>> F.commutes(F)
False
expand()[source]#

Decompose self into a list of construction functors.

Note

The default is to return the list only containing self.

EXAMPLES:

sage: F = QQ.construction()[0]
sage: F.expand()
[FractionField]
sage: Q = ZZ.quo(2).construction()[0]
sage: Q.expand()
[QuotientFunctor]
sage: P = ZZ['t'].construction()[0]
sage: FP = F*P
sage: FP.expand()
[FractionField, Poly[t]]

>>> from sage.all import *
>>> F = QQ.construction()[Integer(0)]
>>> F.expand()
[FractionField]
>>> Q = ZZ.quo(Integer(2)).construction()[Integer(0)]
>>> Q.expand()
[QuotientFunctor]
>>> P = ZZ['t'].construction()[Integer(0)]
>>> FP = F*P
>>> FP.expand()
[FractionField, Poly[t]]
merge(other)[source]#

Merge self with another construction functor, or return None.

Note

The default is to merge only if the two functors coincide. But this may be overloaded for subclasses, such as the quotient functor.

EXAMPLES:

sage: F = QQ.construction()[0]
sage: P = ZZ['t'].construction()[0]
sage: F.merge(F)
FractionField
sage: F.merge(P)
sage: P.merge(F)
sage: P.merge(P)
Poly[t]

>>> from sage.all import *
>>> F = QQ.construction()[Integer(0)]
>>> P = ZZ['t'].construction()[Integer(0)]
>>> F.merge(F)
FractionField
>>> F.merge(P)
>>> P.merge(F)
>>> P.merge(P)
Poly[t]
pushout(other)[source]#

Composition of two construction functors, ordered by their ranks.

Note

  • This method seems not to be used in the coercion model.

  • By default, the functor with smaller rank is applied first.

class sage.categories.pushout.EquivariantSubobjectConstructionFunctor(S, action=<built-in function mul>, side='left', other_action=None, other_side='left')[source]#

Bases: ConstructionFunctor

Constructor for subobjects invariant or equivariant under given semigroup actions.

Let \(S\) be a semigroup that - acts on a parent \(X\) as \(s \cdot x\) (action, side='left') or - acts on \(X\) as \(x \cdot s\) (action, side='right'), and (possibly trivially) - acts on \(X\) as \(s * x\) (other_action, other_side='left') or - acts on \(X\) as \(x * s\) (other_action, other_side='right').

The \(S\)-equivariant subobject is the subobject

\[X^S := \{x \in X : s \cdot x = s * x,\, \forall s \in S \}\]

when side = other_side = 'left' and mutatis mutandis for the other values of side and other_side.

When other_action is trivial, \(X^S\) is called the \(S\)-invariant subobject.

EXAMPLES:

Monoterm symmetries of a tensor, here only for matrices: row (index 0), column (index 1); the order of the extra element 2 in a permutation determines whether it is a symmetry or an antisymmetry:

sage: # needs sage.groups sage.modules
sage: GSym01 = PermutationGroup([[(0,1),(2,),(3,)]]); GSym01
Permutation Group with generators [(0,1)]
sage: GASym01 = PermutationGroup([[(0,1),(2,3)]]); GASym01
Permutation Group with generators [(0,1)(2,3)]
sage: from sage.categories.action import Action
sage: from sage.structure.element import Matrix
sage: class TensorIndexAction(Action):
....:     def _act_(self, g, x):
....:         if isinstance(x, Matrix):
....:             if g(0) == 1:
....:                 if g(2) == 2:
....:                     return x.transpose()
....:                 else:
....:                     return -x.transpose()
....:             else:
....:                 return x
....:         raise NotImplementedError
sage: M = matrix([[1, 2], [3, 4]]); M
[1 2]
[3 4]
sage: GSym01_action = TensorIndexAction(GSym01, M.parent())
sage: GASym01_action = TensorIndexAction(GASym01, M.parent())
sage: GSym01_action.act(GSym01.0, M)
[1 3]
[2 4]
sage: GASym01_action.act(GASym01.0, M)
[-1 -3]
[-2 -4]
sage: Sym01 = M.parent().invariant_module(GSym01, action=GSym01_action); Sym01
(Permutation Group with generators [(0,1)])-invariant submodule
 of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: list(Sym01.basis())
[B[0], B[1], B[2]]
sage: list(Sym01.basis().map(Sym01.lift))
[
[1 0]  [0 1]  [0 0]
[0 0], [1 0], [0 1]
]
sage: ASym01 = M.parent().invariant_module(GASym01, action=GASym01_action)
sage: ASym01
(Permutation Group with generators [(0,1)(2,3)])-invariant submodule
 of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: list(ASym01.basis())
[B[0]]
sage: list(ASym01.basis().map(ASym01.lift))
[
[ 0  1]
[-1  0]
]
sage: from sage.categories.pushout import pushout
sage: pushout(Sym01, QQ)
(Permutation Group with generators [(0,1)])-invariant submodule
 of Full MatrixSpace of 2 by 2 dense matrices over Rational Field

>>> from sage.all import *
>>> # needs sage.groups sage.modules
>>> GSym01 = PermutationGroup([[(Integer(0),Integer(1)),(Integer(2),),(Integer(3),)]]); GSym01
Permutation Group with generators [(0,1)]
>>> GASym01 = PermutationGroup([[(Integer(0),Integer(1)),(Integer(2),Integer(3))]]); GASym01
Permutation Group with generators [(0,1)(2,3)]
>>> from sage.categories.action import Action
>>> from sage.structure.element import Matrix
>>> class TensorIndexAction(Action):
...     def _act_(self, g, x):
...         if isinstance(x, Matrix):
...             if g(Integer(0)) == Integer(1):
...                 if g(Integer(2)) == Integer(2):
...                     return x.transpose()
...                 else:
...                     return -x.transpose()
...             else:
...                 return x
...         raise NotImplementedError
>>> M = matrix([[Integer(1), Integer(2)], [Integer(3), Integer(4)]]); M
[1 2]
[3 4]
>>> GSym01_action = TensorIndexAction(GSym01, M.parent())
>>> GASym01_action = TensorIndexAction(GASym01, M.parent())
>>> GSym01_action.act(GSym01.gen(0), M)
[1 3]
[2 4]
>>> GASym01_action.act(GASym01.gen(0), M)
[-1 -3]
[-2 -4]
>>> Sym01 = M.parent().invariant_module(GSym01, action=GSym01_action); Sym01
(Permutation Group with generators [(0,1)])-invariant submodule
 of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
>>> list(Sym01.basis())
[B[0], B[1], B[2]]
>>> list(Sym01.basis().map(Sym01.lift))
[
[1 0]  [0 1]  [0 0]
[0 0], [1 0], [0 1]
]
>>> ASym01 = M.parent().invariant_module(GASym01, action=GASym01_action)
>>> ASym01
(Permutation Group with generators [(0,1)(2,3)])-invariant submodule
 of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
>>> list(ASym01.basis())
[B[0]]
>>> list(ASym01.basis().map(ASym01.lift))
[
[ 0  1]
[-1  0]
]
>>> from sage.categories.pushout import pushout
>>> pushout(Sym01, QQ)
(Permutation Group with generators [(0,1)])-invariant submodule
 of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
class sage.categories.pushout.FractionField[source]#

Bases: ConstructionFunctor

Construction functor for fraction fields.

EXAMPLES:

sage: F = QQ.construction()[0]
sage: F
FractionField
sage: F.domain()
Category of integral domains
sage: F.codomain()
Category of fields
sage: F(GF(5)) is GF(5)
True
sage: F(ZZ['t'])
Fraction Field of Univariate Polynomial Ring in t over Integer Ring
sage: P.<x,y> = QQ[]
sage: f = P.hom([x+2*y,3*x-y],P)
sage: F(f)
Ring endomorphism of
 Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field
  Defn: x |--> x + 2*y
        y |--> 3*x - y
sage: F(f)(1/x)
1/(x + 2*y)
sage: F == loads(dumps(F))
True

>>> from sage.all import *
>>> F = QQ.construction()[Integer(0)]
>>> F
FractionField
>>> F.domain()
Category of integral domains
>>> F.codomain()
Category of fields
>>> F(GF(Integer(5))) is GF(Integer(5))
True
>>> F(ZZ['t'])
Fraction Field of Univariate Polynomial Ring in t over Integer Ring
>>> P = QQ['x, y']; (x, y,) = P._first_ngens(2)
>>> f = P.hom([x+Integer(2)*y,Integer(3)*x-y],P)
>>> F(f)
Ring endomorphism of
 Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field
  Defn: x |--> x + 2*y
        y |--> 3*x - y
>>> F(f)(Integer(1)/x)
1/(x + 2*y)
>>> F == loads(dumps(F))
True
rank = 5#
class sage.categories.pushout.IdentityConstructionFunctor[source]#

Bases: ConstructionFunctor

A construction functor that is the identity functor.

rank = -100#
class sage.categories.pushout.InfinitePolynomialFunctor(gens, order, implementation)[source]#

Bases: ConstructionFunctor

A Construction Functor for Infinite Polynomial Rings (see infinite_polynomial_ring).

AUTHOR:

– Simon King

This construction functor is used to provide uniqueness of infinite polynomial rings as parent structures. As usual, the construction functor allows for constructing pushouts.

Another purpose is to avoid name conflicts of variables of the to-be-constructed infinite polynomial ring with variables of the base ring, and moreover to keep the internal structure of an Infinite Polynomial Ring as simple as possible: If variables \(v_1,...,v_n\) of the given base ring generate an ordered sub-monoid of the monomials of the ambient Infinite Polynomial Ring, then they are removed from the base ring and merged with the generators of the ambient ring. However, if the orders don’t match, an error is raised, since there was a name conflict without merging.

EXAMPLES:

sage: A.<a,b> = InfinitePolynomialRing(ZZ['t'])
sage: A.construction()
[InfPoly{[a,b], "lex", "dense"},
 Univariate Polynomial Ring in t over Integer Ring]
sage: type(_[0])
<class 'sage.categories.pushout.InfinitePolynomialFunctor'>
sage: B.<x,y,a_3,a_1> = PolynomialRing(QQ, order='lex')
sage: B.construction()
(MPoly[x,y,a_3,a_1], Rational Field)
sage: A.construction()[0] * B.construction()[0]
InfPoly{[a,b], "lex", "dense"}(MPoly[x,y](...))

>>> from sage.all import *
>>> A = InfinitePolynomialRing(ZZ['t'], names=('a', 'b',)); (a, b,) = A._first_ngens(2)
>>> A.construction()
[InfPoly{[a,b], "lex", "dense"},
 Univariate Polynomial Ring in t over Integer Ring]
>>> type(_[Integer(0)])
<class 'sage.categories.pushout.InfinitePolynomialFunctor'>
>>> B = PolynomialRing(QQ, order='lex', names=('x', 'y', 'a_3', 'a_1',)); (x, y, a_3, a_1,) = B._first_ngens(4)
>>> B.construction()
(MPoly[x,y,a_3,a_1], Rational Field)
>>> A.construction()[Integer(0)] * B.construction()[Integer(0)]
InfPoly{[a,b], "lex", "dense"}(MPoly[x,y](...))

Apparently the variables \(a_1,a_3\) of the polynomial ring are merged with the variables \(a_0, a_1, a_2, ...\) of the infinite polynomial ring; indeed, they form an ordered sub-structure. However, if the polynomial ring was given a different ordering, merging would not be allowed, resulting in a name conflict:

sage: R = PolynomialRing(QQ, names=['x','y','a_3','a_1'])
sage: A.construction()[0] * R.construction()[0]
Traceback (most recent call last):
...
CoercionException: Incompatible term orders lex, degrevlex

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=['x','y','a_3','a_1'])
>>> A.construction()[Integer(0)] * R.construction()[Integer(0)]
Traceback (most recent call last):
...
CoercionException: Incompatible term orders lex, degrevlex

In an infinite polynomial ring with generator \(a_\ast\), the variable \(a_3\) will always be greater than the variable \(a_1\). Hence, the orders are incompatible in the next example as well:

sage: R = PolynomialRing(QQ, names=['x','y','a_1','a_3'], order='lex')
sage: A.construction()[0] * R.construction()[0]
Traceback (most recent call last):
...
CoercionException: Overlapping variables (('a', 'b'),['a_1', 'a_3'])
are incompatible

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=['x','y','a_1','a_3'], order='lex')
>>> A.construction()[Integer(0)] * R.construction()[Integer(0)]
Traceback (most recent call last):
...
CoercionException: Overlapping variables (('a', 'b'),['a_1', 'a_3'])
are incompatible

Another requirement is that after merging the order of the remaining variables must be unique. This is not the case in the following example, since it is not clear whether the variables \(x,y\) should be greater or smaller than the variables \(b_\ast\):

sage: R = PolynomialRing(QQ, names=['a_3','a_1','x','y'], order='lex')
sage: A.construction()[0] * R.construction()[0]
Traceback (most recent call last):
...
CoercionException: Overlapping variables (('a', 'b'),['a_3', 'a_1'])
are incompatible

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=['a_3','a_1','x','y'], order='lex')
>>> A.construction()[Integer(0)] * R.construction()[Integer(0)]
Traceback (most recent call last):
...
CoercionException: Overlapping variables (('a', 'b'),['a_3', 'a_1'])
are incompatible

Since the construction functors are actually used to construct infinite polynomial rings, the following result is no surprise:

sage: C.<a,b> = InfinitePolynomialRing(B); C
Infinite polynomial ring in a, b
 over Multivariate Polynomial Ring in x, y over Rational Field

>>> from sage.all import *
>>> C = InfinitePolynomialRing(B, names=('a', 'b',)); (a, b,) = C._first_ngens(2); C
Infinite polynomial ring in a, b
 over Multivariate Polynomial Ring in x, y over Rational Field

There is also an overlap in the next example:

sage: X.<w,x,y> = InfinitePolynomialRing(ZZ)
sage: Y.<x,y,z> = InfinitePolynomialRing(QQ)

>>> from sage.all import *
>>> X = InfinitePolynomialRing(ZZ, names=('w', 'x', 'y',)); (w, x, y,) = X._first_ngens(3)
>>> Y = InfinitePolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = Y._first_ngens(3)

\(X\) and \(Y\) have an overlapping generators \(x_\ast, y_\ast\). Since the default lexicographic order is used in both rings, it gives rise to isomorphic sub-monoids in both \(X\) and \(Y\). They are merged in the pushout, which also yields a common parent for doing arithmetic:

sage: P = sage.categories.pushout.pushout(Y,X); P
Infinite polynomial ring in w, x, y, z over Rational Field
sage: w[2]+z[3]
w_2 + z_3
sage: _.parent() is P
True

>>> from sage.all import *
>>> P = sage.categories.pushout.pushout(Y,X); P
Infinite polynomial ring in w, x, y, z over Rational Field
>>> w[Integer(2)]+z[Integer(3)]
w_2 + z_3
>>> _.parent() is P
True
expand()[source]#

Decompose the functor \(F\) into sub-functors, whose product returns \(F\).

EXAMPLES:

sage: A = InfinitePolynomialRing(QQ, ['x','y'], order='degrevlex')
sage: F = A.construction()[0]; F
InfPoly{[x,y], "degrevlex", "dense"}
sage: F.expand()
[InfPoly{[y], "degrevlex", "dense"}, InfPoly{[x], "degrevlex", "dense"}]
sage: A = InfinitePolynomialRing(QQ, ['x','y','z'], order='degrevlex')
sage: F = A.construction()[0]; F
InfPoly{[x,y,z], "degrevlex", "dense"}
sage: F.expand()
[InfPoly{[z], "degrevlex", "dense"},
 InfPoly{[y], "degrevlex", "dense"},
 InfPoly{[x], "degrevlex", "dense"}]
sage: prod(F.expand())==F
True

>>> from sage.all import *
>>> A = InfinitePolynomialRing(QQ, ['x','y'], order='degrevlex')
>>> F = A.construction()[Integer(0)]; F
InfPoly{[x,y], "degrevlex", "dense"}
>>> F.expand()
[InfPoly{[y], "degrevlex", "dense"}, InfPoly{[x], "degrevlex", "dense"}]
>>> A = InfinitePolynomialRing(QQ, ['x','y','z'], order='degrevlex')
>>> F = A.construction()[Integer(0)]; F
InfPoly{[x,y,z], "degrevlex", "dense"}
>>> F.expand()
[InfPoly{[z], "degrevlex", "dense"},
 InfPoly{[y], "degrevlex", "dense"},
 InfPoly{[x], "degrevlex", "dense"}]
>>> prod(F.expand())==F
True
merge(other)[source]#

Merge two construction functors of infinite polynomial rings, regardless of monomial order and implementation.

The purpose is to have a pushout (and thus, arithmetic) even in cases when the parents are isomorphic as rings, but not as ordered rings.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ, implementation='sparse')
sage: Y.<x,y> = InfinitePolynomialRing(QQ, order='degrevlex')
sage: X.construction()
[InfPoly{[x,y], "lex", "sparse"}, Rational Field]
sage: Y.construction()
[InfPoly{[x,y], "degrevlex", "dense"}, Rational Field]
sage: Y.construction()[0].merge(Y.construction()[0])
InfPoly{[x,y], "degrevlex", "dense"}
sage: y[3] + X(x[2])
x_2 + y_3
sage: _.parent().construction()
[InfPoly{[x,y], "degrevlex", "dense"}, Rational Field]

>>> from sage.all import *
>>> X = InfinitePolynomialRing(QQ, implementation='sparse', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> Y = InfinitePolynomialRing(QQ, order='degrevlex', names=('x', 'y',)); (x, y,) = Y._first_ngens(2)
>>> X.construction()
[InfPoly{[x,y], "lex", "sparse"}, Rational Field]
>>> Y.construction()
[InfPoly{[x,y], "degrevlex", "dense"}, Rational Field]
>>> Y.construction()[Integer(0)].merge(Y.construction()[Integer(0)])
InfPoly{[x,y], "degrevlex", "dense"}
>>> y[Integer(3)] + X(x[Integer(2)])
x_2 + y_3
>>> _.parent().construction()
[InfPoly{[x,y], "degrevlex", "dense"}, Rational Field]
rank = 9.5#
class sage.categories.pushout.LaurentPolynomialFunctor(var, multi_variate=False)[source]#

Bases: ConstructionFunctor

Construction functor for Laurent polynomial rings.

EXAMPLES:

sage: L.<t> = LaurentPolynomialRing(ZZ)
sage: F = L.construction()[0]
sage: F
LaurentPolynomialFunctor
sage: F(QQ)
Univariate Laurent Polynomial Ring in t over Rational Field
sage: K.<x> = LaurentPolynomialRing(ZZ)
sage: F(K)
Univariate Laurent Polynomial Ring in t
 over Univariate Laurent Polynomial Ring in x over Integer Ring
sage: P.<x,y> = ZZ[]
sage: f = P.hom([x + 2*y, 3*x - y],P)
sage: F(f)
Ring endomorphism of Univariate Laurent Polynomial Ring in t
 over Multivariate Polynomial Ring in x, y over Integer Ring
  Defn: Induced from base ring by
        Ring endomorphism of Multivariate Polynomial Ring in x, y over Integer Ring
          Defn: x |--> x + 2*y
                y |--> 3*x - y
sage: F(f)(x*F(P).gen()^-2 + y*F(P).gen()^3)
(x + 2*y)*t^-2 + (3*x - y)*t^3

>>> from sage.all import *
>>> L = LaurentPolynomialRing(ZZ, names=('t',)); (t,) = L._first_ngens(1)
>>> F = L.construction()[Integer(0)]
>>> F
LaurentPolynomialFunctor
>>> F(QQ)
Univariate Laurent Polynomial Ring in t over Rational Field
>>> K = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = K._first_ngens(1)
>>> F(K)
Univariate Laurent Polynomial Ring in t
 over Univariate Laurent Polynomial Ring in x over Integer Ring
>>> P = ZZ['x, y']; (x, y,) = P._first_ngens(2)
>>> f = P.hom([x + Integer(2)*y, Integer(3)*x - y],P)
>>> F(f)
Ring endomorphism of Univariate Laurent Polynomial Ring in t
 over Multivariate Polynomial Ring in x, y over Integer Ring
  Defn: Induced from base ring by
        Ring endomorphism of Multivariate Polynomial Ring in x, y over Integer Ring
          Defn: x |--> x + 2*y
                y |--> 3*x - y
>>> F(f)(x*F(P).gen()**-Integer(2) + y*F(P).gen()**Integer(3))
(x + 2*y)*t^-2 + (3*x - y)*t^3
merge(other)[source]#

Two Laurent polynomial construction functors merge if the variable names coincide.

The result is multivariate if one of the arguments is multivariate.

EXAMPLES:

sage: from sage.categories.pushout import LaurentPolynomialFunctor
sage: F1 = LaurentPolynomialFunctor('t')
sage: F2 = LaurentPolynomialFunctor('t', multi_variate=True)
sage: F1.merge(F2)
LaurentPolynomialFunctor
sage: F1.merge(F2)(LaurentPolynomialRing(GF(2), 'a'))                       # needs sage.modules
Multivariate Laurent Polynomial Ring in a, t over Finite Field of size 2
sage: F1.merge(F1)(LaurentPolynomialRing(GF(2), 'a'))
Univariate Laurent Polynomial Ring in t over
 Univariate Laurent Polynomial Ring in a over Finite Field of size 2

>>> from sage.all import *
>>> from sage.categories.pushout import LaurentPolynomialFunctor
>>> F1 = LaurentPolynomialFunctor('t')
>>> F2 = LaurentPolynomialFunctor('t', multi_variate=True)
>>> F1.merge(F2)
LaurentPolynomialFunctor
>>> F1.merge(F2)(LaurentPolynomialRing(GF(Integer(2)), 'a'))                       # needs sage.modules
Multivariate Laurent Polynomial Ring in a, t over Finite Field of size 2
>>> F1.merge(F1)(LaurentPolynomialRing(GF(Integer(2)), 'a'))
Univariate Laurent Polynomial Ring in t over
 Univariate Laurent Polynomial Ring in a over Finite Field of size 2
rank = 9#
class sage.categories.pushout.MatrixFunctor(nrows, ncols, is_sparse=False)[source]#

Bases: ConstructionFunctor

A construction functor for matrices over rings.

EXAMPLES:

sage: # needs sage.modules
sage: MS = MatrixSpace(ZZ, 2, 3)
sage: F = MS.construction()[0]; F
MatrixFunctor
sage: MS = MatrixSpace(ZZ, 2)
sage: F = MS.construction()[0]; F
MatrixFunctor
sage: P.<x,y> = QQ[]
sage: R = F(P); R
Full MatrixSpace of 2 by 2 dense matrices
 over Multivariate Polynomial Ring in x, y over Rational Field
sage: f = P.hom([x+y, x-y], P); F(f)
Ring endomorphism
 of Full MatrixSpace of 2 by 2 dense matrices
  over Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Induced from base ring by
        Ring endomorphism
         of Multivariate Polynomial Ring in x, y over Rational Field
          Defn: x |--> x + y
                y |--> x - y
sage: M = R([x, y, x*y, x + y])
sage: F(f)(M)
[    x + y     x - y]
[x^2 - y^2       2*x]

>>> from sage.all import *
>>> # needs sage.modules
>>> MS = MatrixSpace(ZZ, Integer(2), Integer(3))
>>> F = MS.construction()[Integer(0)]; F
MatrixFunctor
>>> MS = MatrixSpace(ZZ, Integer(2))
>>> F = MS.construction()[Integer(0)]; F
MatrixFunctor
>>> P = QQ['x, y']; (x, y,) = P._first_ngens(2)
>>> R = F(P); R
Full MatrixSpace of 2 by 2 dense matrices
 over Multivariate Polynomial Ring in x, y over Rational Field
>>> f = P.hom([x+y, x-y], P); F(f)
Ring endomorphism
 of Full MatrixSpace of 2 by 2 dense matrices
  over Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Induced from base ring by
        Ring endomorphism
         of Multivariate Polynomial Ring in x, y over Rational Field
          Defn: x |--> x + y
                y |--> x - y
>>> M = R([x, y, x*y, x + y])
>>> F(f)(M)
[    x + y     x - y]
[x^2 - y^2       2*x]
merge(other)[source]#

Merging is only happening if both functors are matrix functors of the same dimension.

The result is sparse if and only if both given functors are sparse.

EXAMPLES:

sage: # needs sage.modules
sage: F1 = MatrixSpace(ZZ, 2, 2).construction()[0]
sage: F2 = MatrixSpace(ZZ, 2, 3).construction()[0]
sage: F3 = MatrixSpace(ZZ, 2, 2, sparse=True).construction()[0]
sage: F1.merge(F2)
sage: F1.merge(F3)
MatrixFunctor
sage: F13 = F1.merge(F3)
sage: F13.is_sparse
False
sage: F1.is_sparse
False
sage: F3.is_sparse
True
sage: F3.merge(F3).is_sparse
True

>>> from sage.all import *
>>> # needs sage.modules
>>> F1 = MatrixSpace(ZZ, Integer(2), Integer(2)).construction()[Integer(0)]
>>> F2 = MatrixSpace(ZZ, Integer(2), Integer(3)).construction()[Integer(0)]
>>> F3 = MatrixSpace(ZZ, Integer(2), Integer(2), sparse=True).construction()[Integer(0)]
>>> F1.merge(F2)
>>> F1.merge(F3)
MatrixFunctor
>>> F13 = F1.merge(F3)
>>> F13.is_sparse
False
>>> F1.is_sparse
False
>>> F3.is_sparse
True
>>> F3.merge(F3).is_sparse
True
rank = 10#
class sage.categories.pushout.MultiPolynomialFunctor(vars, term_order)[source]#

Bases: ConstructionFunctor

A constructor for multivariate polynomial rings.

EXAMPLES:

sage: P.<x,y> = ZZ[]
sage: F = P.construction()[0]; F
MPoly[x,y]
sage: A.<a,b> = GF(5)[]
sage: F(A)
Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
sage: f = A.hom([a+b, a-b], A)
sage: F(f)
Ring endomorphism of Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
  Defn: Induced from base ring by
        Ring endomorphism of Multivariate Polynomial Ring in a, b over Finite Field of size 5
          Defn: a |--> a + b
                b |--> a - b
sage: F(f)(F(A)(x)*a)
(a + b)*x

>>> from sage.all import *
>>> P = ZZ['x, y']; (x, y,) = P._first_ngens(2)
>>> F = P.construction()[Integer(0)]; F
MPoly[x,y]
>>> A = GF(Integer(5))['a, b']; (a, b,) = A._first_ngens(2)
>>> F(A)
Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
>>> f = A.hom([a+b, a-b], A)
>>> F(f)
Ring endomorphism of Multivariate Polynomial Ring in x, y
 over Multivariate Polynomial Ring in a, b over Finite Field of size 5
  Defn: Induced from base ring by
        Ring endomorphism of Multivariate Polynomial Ring in a, b over Finite Field of size 5
          Defn: a |--> a + b
                b |--> a - b
>>> F(f)(F(A)(x)*a)
(a + b)*x
expand()[source]#

Decompose self into a list of construction functors.

EXAMPLES:

sage: F = QQ['x,y,z,t'].construction()[0]; F
MPoly[x,y,z,t]
sage: F.expand()
[MPoly[t], MPoly[z], MPoly[y], MPoly[x]]

>>> from sage.all import *
>>> F = QQ['x,y,z,t'].construction()[Integer(0)]; F
MPoly[x,y,z,t]
>>> F.expand()
[MPoly[t], MPoly[z], MPoly[y], MPoly[x]]

Now an actual use case:

sage: R.<x,y,z> = ZZ[]
sage: S.<z,t> = QQ[]
sage: x+t
x + t
sage: parent(x+t)
Multivariate Polynomial Ring in x, y, z, t over Rational Field
sage: T.<y,s> = QQ[]
sage: x + s
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Multivariate Polynomial Ring in x, y, z over Integer Ring' and
'Multivariate Polynomial Ring in y, s over Rational Field'
sage: R = PolynomialRing(ZZ, 'x', 50)
sage: S = PolynomialRing(GF(5), 'x', 20)
sage: R.gen(0) + S.gen(0)
2*x0

>>> from sage.all import *
>>> R = ZZ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> S = QQ['z, t']; (z, t,) = S._first_ngens(2)
>>> x+t
x + t
>>> parent(x+t)
Multivariate Polynomial Ring in x, y, z, t over Rational Field
>>> T = QQ['y, s']; (y, s,) = T._first_ngens(2)
>>> x + s
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Multivariate Polynomial Ring in x, y, z over Integer Ring' and
'Multivariate Polynomial Ring in y, s over Rational Field'
>>> R = PolynomialRing(ZZ, 'x', Integer(50))
>>> S = PolynomialRing(GF(Integer(5)), 'x', Integer(20))
>>> R.gen(Integer(0)) + S.gen(Integer(0))
2*x0
merge(other)[source]#

Merge self with another construction functor, or return None.

EXAMPLES:

sage: F = sage.categories.pushout.MultiPolynomialFunctor(['x','y'], None)
sage: G = sage.categories.pushout.MultiPolynomialFunctor(['t'], None)
sage: F.merge(G) is None
True
sage: F.merge(F)
MPoly[x,y]

>>> from sage.all import *
>>> F = sage.categories.pushout.MultiPolynomialFunctor(['x','y'], None)
>>> G = sage.categories.pushout.MultiPolynomialFunctor(['t'], None)
>>> F.merge(G) is None
True
>>> F.merge(F)
MPoly[x,y]
rank = 9#
class sage.categories.pushout.MultivariateConstructionFunctor[source]#

Bases: ConstructionFunctor

An abstract base class for functors that take multiple inputs (e.g. Cartesian products).

common_base(other_functor, self_bases, other_bases)[source]#

This function is called by pushout() when no common parent is found in the construction tower.

INPUT:

  • other_functor – a construction functor.

  • self_bases – the arguments passed to this functor.

  • other_bases – the arguments passed to the functor other_functor.

OUTPUT:

A parent.

If no common base is found a sage.structure.coerce_exceptions.CoercionException is raised.

Note

Overload this function in derived class, see e.g. MultivariateConstructionFunctor.

class sage.categories.pushout.PermutationGroupFunctor(gens, domain)[source]#

Bases: ConstructionFunctor

EXAMPLES:

sage: from sage.categories.pushout import PermutationGroupFunctor
sage: PF = PermutationGroupFunctor([PermutationGroupElement([(1,2)])],      # needs sage.groups
....:                              [1,2]); PF
PermutationGroupFunctor[(1,2)]

>>> from sage.all import *
>>> from sage.categories.pushout import PermutationGroupFunctor
>>> PF = PermutationGroupFunctor([PermutationGroupElement([(Integer(1),Integer(2))])],      # needs sage.groups
...                              [Integer(1),Integer(2)]); PF
PermutationGroupFunctor[(1,2)]
gens()[source]#

EXAMPLES:

sage: P1 = PermutationGroup([[(1,2)]])                                      # needs sage.groups
sage: PF, P = P1.construction()                                             # needs sage.groups
sage: PF.gens()                                                             # needs sage.groups
((1,2),)

>>> from sage.all import *
>>> P1 = PermutationGroup([[(Integer(1),Integer(2))]])                                      # needs sage.groups
>>> PF, P = P1.construction()                                             # needs sage.groups
>>> PF.gens()                                                             # needs sage.groups
((1,2),)
merge(other)[source]#

Merge self with another construction functor, or return None.

EXAMPLES:

sage: # needs sage.groups
sage: P1 = PermutationGroup([[(1,2)]])
sage: PF1, P = P1.construction()
sage: P2 = PermutationGroup([[(1,3)]])
sage: PF2, P = P2.construction()
sage: PF1.merge(PF2)
PermutationGroupFunctor[(1,2), (1,3)]

>>> from sage.all import *
>>> # needs sage.groups
>>> P1 = PermutationGroup([[(Integer(1),Integer(2))]])
>>> PF1, P = P1.construction()
>>> P2 = PermutationGroup([[(Integer(1),Integer(3))]])
>>> PF2, P = P2.construction()
>>> PF1.merge(PF2)
PermutationGroupFunctor[(1,2), (1,3)]
rank = 10#
class sage.categories.pushout.PolynomialFunctor(var, multi_variate=False, sparse=False, implementation=None)[source]#

Bases: ConstructionFunctor

Construction functor for univariate polynomial rings.

EXAMPLES:

sage: P = ZZ['t'].construction()[0]
sage: P(GF(3))
Univariate Polynomial Ring in t over Finite Field of size 3
sage: P == loads(dumps(P))
True
sage: R.<x,y> = GF(5)[]
sage: f = R.hom([x + 2*y, 3*x - y], R)
sage: P(f)((x+y) * P(R).0)
(-x + y)*t

>>> from sage.all import *
>>> P = ZZ['t'].construction()[Integer(0)]
>>> P(GF(Integer(3)))
Univariate Polynomial Ring in t over Finite Field of size 3
>>> P == loads(dumps(P))
True
>>> R = GF(Integer(5))['x, y']; (x, y,) = R._first_ngens(2)
>>> f = R.hom([x + Integer(2)*y, Integer(3)*x - y], R)
>>> P(f)((x+y) * P(R).gen(0))
(-x + y)*t

By Issue #9944, the construction functor distinguishes sparse and dense polynomial rings. Before, the following example failed:

sage: R.<x> = PolynomialRing(GF(5), sparse=True)
sage: F, B = R.construction()
sage: F(B) is R
True
sage: S.<x> = PolynomialRing(ZZ)
sage: R.has_coerce_map_from(S)
False
sage: S.has_coerce_map_from(R)
False
sage: S.0 + R.0
2*x
sage: (S.0 + R.0).parent()
Univariate Polynomial Ring in x over Finite Field of size 5
sage: (S.0 + R.0).parent().is_sparse()
False

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(5)), sparse=True, names=('x',)); (x,) = R._first_ngens(1)
>>> F, B = R.construction()
>>> F(B) is R
True
>>> S = PolynomialRing(ZZ, names=('x',)); (x,) = S._first_ngens(1)
>>> R.has_coerce_map_from(S)
False
>>> S.has_coerce_map_from(R)
False
>>> S.gen(0) + R.gen(0)
2*x
>>> (S.gen(0) + R.gen(0)).parent()
Univariate Polynomial Ring in x over Finite Field of size 5
>>> (S.gen(0) + R.gen(0)).parent().is_sparse()
False
merge(other)[source]#

Merge self with another construction functor, or return None.

Note

Internally, the merging is delegated to the merging of multipolynomial construction functors. But in effect, this does the same as the default implementation, that returns None unless the to-be-merged functors coincide.

EXAMPLES:

sage: P = ZZ['x'].construction()[0]
sage: Q = ZZ['y','x'].construction()[0]
sage: P.merge(Q)
sage: P.merge(P) is P
True

>>> from sage.all import *
>>> P = ZZ['x'].construction()[Integer(0)]
>>> Q = ZZ['y','x'].construction()[Integer(0)]
>>> P.merge(Q)
>>> P.merge(P) is P
True
rank = 9#
class sage.categories.pushout.QuotientFunctor(I, names=None, as_field=False, domain=None, codomain=None, **kwds)[source]#

Bases: ConstructionFunctor

Construction functor for quotient rings.

Note

The functor keeps track of variable names. Optionally, it may keep track of additional properties of the quotient, such as its category or its implementation.

EXAMPLES:

sage: P.<x,y> = ZZ[]
sage: Q = P.quo([x^2 + y^2] * P)
sage: F = Q.construction()[0]
sage: F(QQ['x','y'])
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
 by the ideal (x^2 + y^2)
sage: F(QQ['x','y']) == QQ['x','y'].quo([x^2 + y^2] * QQ['x','y'])
True
sage: F(QQ['x','y','z'])
Traceback (most recent call last):
...
CoercionException: Cannot apply this quotient functor to
 Multivariate Polynomial Ring in x, y, z over Rational Field
sage: F(QQ['y','z'])                                                            # needs sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: Could not find a mapping of the passed element to this ring.

>>> from sage.all import *
>>> P = ZZ['x, y']; (x, y,) = P._first_ngens(2)
>>> Q = P.quo([x**Integer(2) + y**Integer(2)] * P)
>>> F = Q.construction()[Integer(0)]
>>> F(QQ['x','y'])
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
 by the ideal (x^2 + y^2)
>>> F(QQ['x','y']) == QQ['x','y'].quo([x**Integer(2) + y**Integer(2)] * QQ['x','y'])
True
>>> F(QQ['x','y','z'])
Traceback (most recent call last):
...
CoercionException: Cannot apply this quotient functor to
 Multivariate Polynomial Ring in x, y, z over Rational Field
>>> F(QQ['y','z'])                                                            # needs sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: Could not find a mapping of the passed element to this ring.
merge(other)[source]#

Two quotient functors with coinciding names are merged by taking the gcd of their moduli, the meet of their domains, and the join of their codomains.

In particular, if one of the functors being merged knows that the quotient is going to be a field, then the merged functor will return fields as well.

EXAMPLES:

sage: # needs sage.libs.pari
sage: P.<x> = QQ[]
sage: Q1 = P.quo([(x^2+1)^2*(x^2-3)])
sage: Q2 = P.quo([(x^2+1)^2*(x^5+3)])
sage: from sage.categories.pushout import pushout
sage: pushout(Q1,Q2)    # indirect doctest
Univariate Quotient Polynomial Ring in xbar over Rational Field
 with modulus x^4 + 2*x^2 + 1

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> P = QQ['x']; (x,) = P._first_ngens(1)
>>> Q1 = P.quo([(x**Integer(2)+Integer(1))**Integer(2)*(x**Integer(2)-Integer(3))])
>>> Q2 = P.quo([(x**Integer(2)+Integer(1))**Integer(2)*(x**Integer(5)+Integer(3))])
>>> from sage.categories.pushout import pushout
>>> pushout(Q1,Q2)    # indirect doctest
Univariate Quotient Polynomial Ring in xbar over Rational Field
 with modulus x^4 + 2*x^2 + 1

The following was fixed in Issue #8800:

sage: pushout(GF(5), Integers(5))                                           # needs sage.libs.pari
Finite Field of size 5

>>> from sage.all import *
>>> pushout(GF(Integer(5)), Integers(Integer(5)))                                           # needs sage.libs.pari
Finite Field of size 5
rank = 4.5#
class sage.categories.pushout.SubspaceFunctor(basis)[source]#

Bases: ConstructionFunctor

Constructing a subspace of an ambient free module, given by a basis.

Note

This construction functor keeps track of the basis. It can only be applied to free modules into which this basis coerces.

EXAMPLES:

sage: # needs sage.modules
sage: M = ZZ^3
sage: S = M.submodule([(1,2,3), (4,5,6)]); S
Free module of degree 3 and rank 2 over Integer Ring
Echelon basis matrix:
[1 2 3]
[0 3 6]
sage: F = S.construction()[0]
sage: F(GF(2)^3)
Vector space of degree 3 and dimension 2 over Finite Field of size 2
User basis matrix:
[1 0 1]
[0 1 0]

>>> from sage.all import *
>>> # needs sage.modules
>>> M = ZZ**Integer(3)
>>> S = M.submodule([(Integer(1),Integer(2),Integer(3)), (Integer(4),Integer(5),Integer(6))]); S
Free module of degree 3 and rank 2 over Integer Ring
Echelon basis matrix:
[1 2 3]
[0 3 6]
>>> F = S.construction()[Integer(0)]
>>> F(GF(Integer(2))**Integer(3))
Vector space of degree 3 and dimension 2 over Finite Field of size 2
User basis matrix:
[1 0 1]
[0 1 0]
coercion_reversed = True#
merge(other)[source]#

Two Subspace Functors are merged into a construction functor of the sum of two subspaces.

EXAMPLES:

sage: # needs sage.modules
sage: M = GF(5)^3
sage: S1 = M.submodule([(1,2,3),(4,5,6)])
sage: S2 = M.submodule([(2,2,3)])
sage: F1 = S1.construction()[0]
sage: F2 = S2.construction()[0]
sage: F1.merge(F2)
SubspaceFunctor
sage: F1.merge(F2)(GF(5)^3) == S1 + S2
True
sage: F1.merge(F2)(GF(5)['t']^3)
Free module of degree 3 and rank 3
 over Univariate Polynomial Ring in t over Finite Field of size 5
User basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]

>>> from sage.all import *
>>> # needs sage.modules
>>> M = GF(Integer(5))**Integer(3)
>>> S1 = M.submodule([(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5),Integer(6))])
>>> S2 = M.submodule([(Integer(2),Integer(2),Integer(3))])
>>> F1 = S1.construction()[Integer(0)]
>>> F2 = S2.construction()[Integer(0)]
>>> F1.merge(F2)
SubspaceFunctor
>>> F1.merge(F2)(GF(Integer(5))**Integer(3)) == S1 + S2
True
>>> F1.merge(F2)(GF(Integer(5))['t']**Integer(3))
Free module of degree 3 and rank 3
 over Univariate Polynomial Ring in t over Finite Field of size 5
User basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
rank = 11#
class sage.categories.pushout.VectorFunctor(n=None, is_sparse=False, inner_product_matrix=None, *, with_basis='standard', basis_keys=None, name_mapping=None, latex_name_mapping=None)[source]#

Bases: ConstructionFunctor

A construction functor for free modules over commutative rings.

EXAMPLES:

sage: # needs sage.modules
sage: F = (ZZ^3).construction()[0]
sage: F
VectorFunctor
sage: F(GF(2)['t'])                                                             # needs sage.libs.ntl
Ambient free module of rank 3
 over the principal ideal domain Univariate Polynomial Ring in t
  over Finite Field of size 2 (using GF2X)

>>> from sage.all import *
>>> # needs sage.modules
>>> F = (ZZ**Integer(3)).construction()[Integer(0)]
>>> F
VectorFunctor
>>> F(GF(Integer(2))['t'])                                                             # needs sage.libs.ntl
Ambient free module of rank 3
 over the principal ideal domain Univariate Polynomial Ring in t
  over Finite Field of size 2 (using GF2X)
merge(other)[source]#

Two constructors of free modules merge, if the module ranks and the inner products coincide. If both have explicitly given inner product matrices, they must coincide as well.

EXAMPLES:

Two modules without explicitly given inner product allow coercion:

sage: M1 = QQ^3                                                             # needs sage.modules
sage: P.<t> = ZZ[]
sage: M2 = FreeModule(P, 3)                                                 # needs sage.modules
sage: M1([1,1/2,1/3]) + M2([t,t^2+t,3])     # indirect doctest              # needs sage.modules
(t + 1, t^2 + t + 1/2, 10/3)

>>> from sage.all import *
>>> M1 = QQ**Integer(3)                                                             # needs sage.modules
>>> P = ZZ['t']; (t,) = P._first_ngens(1)
>>> M2 = FreeModule(P, Integer(3))                                                 # needs sage.modules
>>> M1([Integer(1),Integer(1)/Integer(2),Integer(1)/Integer(3)]) + M2([t,t**Integer(2)+t,Integer(3)])     # indirect doctest              # needs sage.modules
(t + 1, t^2 + t + 1/2, 10/3)

If only one summand has an explicit inner product, the result will be provided with it:

sage: M3 = FreeModule(P, 3, inner_product_matrix=Matrix(3, 3, range(9)))    # needs sage.modules
sage: M1([1,1/2,1/3]) + M3([t,t^2+t,3])                                     # needs sage.modules
(t + 1, t^2 + t + 1/2, 10/3)
sage: (M1([1,1/2,1/3]) + M3([t,t^2+t,3])).parent().inner_product_matrix()   # needs sage.modules
[0 1 2]
[3 4 5]
[6 7 8]

>>> from sage.all import *
>>> M3 = FreeModule(P, Integer(3), inner_product_matrix=Matrix(Integer(3), Integer(3), range(Integer(9))))    # needs sage.modules
>>> M1([Integer(1),Integer(1)/Integer(2),Integer(1)/Integer(3)]) + M3([t,t**Integer(2)+t,Integer(3)])                                     # needs sage.modules
(t + 1, t^2 + t + 1/2, 10/3)
>>> (M1([Integer(1),Integer(1)/Integer(2),Integer(1)/Integer(3)]) + M3([t,t**Integer(2)+t,Integer(3)])).parent().inner_product_matrix()   # needs sage.modules
[0 1 2]
[3 4 5]
[6 7 8]

If both summands have an explicit inner product (even if it is the standard inner product), then the products must coincide. The only difference between M1 and M4 in the following example is the fact that the default inner product was explicitly requested for M4. It is therefore not possible to coerce with a different inner product:

sage: # needs sage.modules
sage: M4 = FreeModule(QQ, 3, inner_product_matrix=Matrix(3, 3, 1))
sage: M4 == M1
True
sage: M4.inner_product_matrix() == M1.inner_product_matrix()
True
sage: M4([1,1/2,1/3]) + M3([t,t^2+t,3])      # indirect doctest
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ambient quadratic space of dimension 3 over Rational Field
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]' and
'Ambient free quadratic module of rank 3 over the integral domain
Univariate Polynomial Ring in t over Integer Ring
Inner product matrix:
[0 1 2]
[3 4 5]
[6 7 8]'

>>> from sage.all import *
>>> # needs sage.modules
>>> M4 = FreeModule(QQ, Integer(3), inner_product_matrix=Matrix(Integer(3), Integer(3), Integer(1)))
>>> M4 == M1
True
>>> M4.inner_product_matrix() == M1.inner_product_matrix()
True
>>> M4([Integer(1),Integer(1)/Integer(2),Integer(1)/Integer(3)]) + M3([t,t**Integer(2)+t,Integer(3)])      # indirect doctest
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ambient quadratic space of dimension 3 over Rational Field
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]' and
'Ambient free quadratic module of rank 3 over the integral domain
Univariate Polynomial Ring in t over Integer Ring
Inner product matrix:
[0 1 2]
[3 4 5]
[6 7 8]'

Names are removed when they conflict:

sage: # needs sage.modules
sage: from sage.categories.pushout import VectorFunctor, pushout
sage: M_ZZx = FreeModule(ZZ['x'], 4, with_basis=None, name='M_ZZx')
sage: N_ZZx = FreeModule(ZZ['x'], 4, with_basis=None, name='N_ZZx')
sage: pushout(M_ZZx, QQ)
Rank-4 free module M_ZZx_base_ext
 over the Univariate Polynomial Ring in x over Rational Field
sage: pushout(M_ZZx, N_ZZx)
Rank-4 free module
 over the Univariate Polynomial Ring in x over Integer Ring
sage: pushout(pushout(M_ZZx, N_ZZx), QQ)
Rank-4 free module
 over the Univariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> # needs sage.modules
>>> from sage.categories.pushout import VectorFunctor, pushout
>>> M_ZZx = FreeModule(ZZ['x'], Integer(4), with_basis=None, name='M_ZZx')
>>> N_ZZx = FreeModule(ZZ['x'], Integer(4), with_basis=None, name='N_ZZx')
>>> pushout(M_ZZx, QQ)
Rank-4 free module M_ZZx_base_ext
 over the Univariate Polynomial Ring in x over Rational Field
>>> pushout(M_ZZx, N_ZZx)
Rank-4 free module
 over the Univariate Polynomial Ring in x over Integer Ring
>>> pushout(pushout(M_ZZx, N_ZZx), QQ)
Rank-4 free module
 over the Univariate Polynomial Ring in x over Rational Field
rank = 10#
sage.categories.pushout.construction_tower(R)[source]#

An auxiliary function that is used in pushout() and pushout_lattice().

INPUT:

An object

OUTPUT:

A constructive description of the object from scratch, by a list of pairs of a construction functor and an object to which the construction functor is to be applied. The first pair is formed by None and the given object.

EXAMPLES:

sage: from sage.categories.pushout import construction_tower
sage: construction_tower(MatrixSpace(FractionField(QQ['t']), 2))                # needs sage.modules
[(None, Full MatrixSpace of 2 by 2 dense matrices over Fraction Field
         of Univariate Polynomial Ring in t over Rational Field),
 (MatrixFunctor, Fraction Field
                  of Univariate Polynomial Ring in t over Rational Field),
 (FractionField, Univariate Polynomial Ring in t over Rational Field),
 (Poly[t], Rational Field), (FractionField, Integer Ring)]

>>> from sage.all import *
>>> from sage.categories.pushout import construction_tower
>>> construction_tower(MatrixSpace(FractionField(QQ['t']), Integer(2)))                # needs sage.modules
[(None, Full MatrixSpace of 2 by 2 dense matrices over Fraction Field
         of Univariate Polynomial Ring in t over Rational Field),
 (MatrixFunctor, Fraction Field
                  of Univariate Polynomial Ring in t over Rational Field),
 (FractionField, Univariate Polynomial Ring in t over Rational Field),
 (Poly[t], Rational Field), (FractionField, Integer Ring)]
sage.categories.pushout.expand_tower(tower)[source]#

An auxiliary function that is used in pushout().

INPUT:

A construction tower as returned by construction_tower().

OUTPUT:

A new construction tower with all the construction functors expanded.

EXAMPLES:

sage: from sage.categories.pushout import construction_tower, expand_tower
sage: construction_tower(QQ['x,y,z'])
[(None, Multivariate Polynomial Ring in x, y, z over Rational Field),
 (MPoly[x,y,z], Rational Field),
 (FractionField, Integer Ring)]
sage: expand_tower(construction_tower(QQ['x,y,z']))
[(None, Multivariate Polynomial Ring in x, y, z over Rational Field),
 (MPoly[z], Univariate Polynomial Ring in y
             over Univariate Polynomial Ring in x over Rational Field),
 (MPoly[y], Univariate Polynomial Ring in x over Rational Field),
 (MPoly[x], Rational Field),
 (FractionField, Integer Ring)]

>>> from sage.all import *
>>> from sage.categories.pushout import construction_tower, expand_tower
>>> construction_tower(QQ['x,y,z'])
[(None, Multivariate Polynomial Ring in x, y, z over Rational Field),
 (MPoly[x,y,z], Rational Field),
 (FractionField, Integer Ring)]
>>> expand_tower(construction_tower(QQ['x,y,z']))
[(None, Multivariate Polynomial Ring in x, y, z over Rational Field),
 (MPoly[z], Univariate Polynomial Ring in y
             over Univariate Polynomial Ring in x over Rational Field),
 (MPoly[y], Univariate Polynomial Ring in x over Rational Field),
 (MPoly[x], Rational Field),
 (FractionField, Integer Ring)]
sage.categories.pushout.pushout(R, S)[source]#

Given a pair of objects \(R\) and \(S\), try to construct a reasonable object \(Y\) and return maps such that canonically \(R \leftarrow Y \rightarrow S\).

ALGORITHM:

This incorporates the idea of functors discussed at Sage Days 4. Every object \(R\) can be viewed as an initial object and a series of functors (e.g. polynomial, quotient, extension, completion, vector/matrix, etc.). Call the series of increasingly simple objects (with the associated functors) the “tower” of \(R\). The construction method is used to create the tower.

Given two objects \(R\) and \(S\), try to find a common initial object \(Z\). If the towers of \(R\) and \(S\) meet, let \(Z\) be their join. Otherwise, see if the top of one coerces naturally into the other.

Now we have an initial object and two ordered lists of functors to apply. We wish to merge these in an unambiguous order, popping elements off the top of one or the other tower as we apply them to \(Z\).

  • If the functors are of distinct types, there is an absolute ordering given by the rank attribute. Use this.

  • Otherwise:

    • If the tops are equal, we (try to) merge them.

    • If exactly one occurs lower in the other tower, we may unambiguously apply the other (hoping for a later merge).

    • If the tops commute, we can apply either first.

    • Otherwise fail due to ambiguity.

The algorithm assumes by default that when a construction \(F\) is applied to an object \(X\), the object \(F(X)\) admits a coercion map from \(X\). However, the algorithm can also handle the case where \(F(X)\) has a coercion map to \(X\) instead. In this case, the attribute coercion_reversed of the class implementing \(F\) should be set to True.

EXAMPLES:

Here our “towers” are \(R = Complete_7(Frac(\ZZ))\) and \(Frac(Poly_x(\ZZ))\), which give us \(Frac(Poly_x(Complete_7(Frac(\ZZ))))\):

sage: from sage.categories.pushout import pushout
sage: pushout(Qp(7), Frac(ZZ['x']))                                             # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20

>>> from sage.all import *
>>> from sage.categories.pushout import pushout
>>> pushout(Qp(Integer(7)), Frac(ZZ['x']))                                             # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20

Note we get the same thing with

sage: pushout(Zp(7), Frac(QQ['x']))                                             # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20
sage: pushout(Zp(7)['x'], Frac(QQ['x']))                                        # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20

>>> from sage.all import *
>>> pushout(Zp(Integer(7)), Frac(QQ['x']))                                             # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20
>>> pushout(Zp(Integer(7))['x'], Frac(QQ['x']))                                        # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20

Note that polynomial variable ordering must be unambiguously determined.

sage: pushout(ZZ['x,y,z'], QQ['w,z,t'])
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension',
Multivariate Polynomial Ring in x, y, z over Integer Ring,
Multivariate Polynomial Ring in w, z, t over Rational Field)
sage: pushout(ZZ['x,y,z'], QQ['w,x,z,t'])
Multivariate Polynomial Ring in w, x, y, z, t over Rational Field

>>> from sage.all import *
>>> pushout(ZZ['x,y,z'], QQ['w,z,t'])
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension',
Multivariate Polynomial Ring in x, y, z over Integer Ring,
Multivariate Polynomial Ring in w, z, t over Rational Field)
>>> pushout(ZZ['x,y,z'], QQ['w,x,z,t'])
Multivariate Polynomial Ring in w, x, y, z, t over Rational Field

Some other examples:

sage: pushout(Zp(7)['y'], Frac(QQ['t'])['x,y,z'])                               # needs sage.rings.padics
Multivariate Polynomial Ring in x, y, z
 over Fraction Field of Univariate Polynomial Ring in t
  over 7-adic Field with capped relative precision 20
sage: pushout(ZZ['x,y,z'], Frac(ZZ['x'])['y'])
Multivariate Polynomial Ring in y, z
 over Fraction Field of Univariate Polynomial Ring in x over Integer Ring
sage: pushout(MatrixSpace(RDF, 2, 2), Frac(ZZ['x']))                            # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices
 over Fraction Field of Univariate Polynomial Ring in x over Real Double Field
sage: pushout(ZZ, MatrixSpace(ZZ[['x']], 3, 3))                                 # needs sage.modules
Full MatrixSpace of 3 by 3 dense matrices
 over Power Series Ring in x over Integer Ring
sage: pushout(QQ['x,y'], ZZ[['x']])
Univariate Polynomial Ring in y
 over Power Series Ring in x over Rational Field
sage: pushout(Frac(ZZ['x']), QQ[['x']])
Laurent Series Ring in x over Rational Field

>>> from sage.all import *
>>> pushout(Zp(Integer(7))['y'], Frac(QQ['t'])['x,y,z'])                               # needs sage.rings.padics
Multivariate Polynomial Ring in x, y, z
 over Fraction Field of Univariate Polynomial Ring in t
  over 7-adic Field with capped relative precision 20
>>> pushout(ZZ['x,y,z'], Frac(ZZ['x'])['y'])
Multivariate Polynomial Ring in y, z
 over Fraction Field of Univariate Polynomial Ring in x over Integer Ring
>>> pushout(MatrixSpace(RDF, Integer(2), Integer(2)), Frac(ZZ['x']))                            # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices
 over Fraction Field of Univariate Polynomial Ring in x over Real Double Field
>>> pushout(ZZ, MatrixSpace(ZZ[['x']], Integer(3), Integer(3)))                                 # needs sage.modules
Full MatrixSpace of 3 by 3 dense matrices
 over Power Series Ring in x over Integer Ring
>>> pushout(QQ['x,y'], ZZ[['x']])
Univariate Polynomial Ring in y
 over Power Series Ring in x over Rational Field
>>> pushout(Frac(ZZ['x']), QQ[['x']])
Laurent Series Ring in x over Rational Field

A construction with coercion_reversed=True (currently only the SubspaceFunctor construction) is only applied if it leads to a valid coercion:

sage: # needs sage.modules
sage: A = ZZ^2
sage: V = span([[1, 2]], QQ)
sage: P = sage.categories.pushout.pushout(A, V)
sage: P
Vector space of dimension 2 over Rational Field
sage: P.has_coerce_map_from(A)
True

sage: # needs sage.modules
sage: V = (QQ^3).span([[1, 2, 3/4]])
sage: A = ZZ^3
sage: pushout(A, V)
Vector space of dimension 3 over Rational Field
sage: B = A.span([[0, 0, 2/3]])
sage: pushout(B, V)
Vector space of degree 3 and dimension 2 over Rational Field
User basis matrix:
[1 2 0]
[0 0 1]

>>> from sage.all import *
>>> # needs sage.modules
>>> A = ZZ**Integer(2)
>>> V = span([[Integer(1), Integer(2)]], QQ)
>>> P = sage.categories.pushout.pushout(A, V)
>>> P
Vector space of dimension 2 over Rational Field
>>> P.has_coerce_map_from(A)
True

>>> # needs sage.modules
>>> V = (QQ**Integer(3)).span([[Integer(1), Integer(2), Integer(3)/Integer(4)]])
>>> A = ZZ**Integer(3)
>>> pushout(A, V)
Vector space of dimension 3 over Rational Field
>>> B = A.span([[Integer(0), Integer(0), Integer(2)/Integer(3)]])
>>> pushout(B, V)
Vector space of degree 3 and dimension 2 over Rational Field
User basis matrix:
[1 2 0]
[0 0 1]

Some more tests with coercion_reversed=True:

sage: from sage.categories.pushout import ConstructionFunctor
sage: class EvenPolynomialRing(type(QQ['x'])):
....:     def __init__(self, base, var):
....:         super().__init__(base, var)
....:         self.register_embedding(base[var])
....:     def __repr__(self):
....:         return "Even Power " + super().__repr__()
....:     def construction(self):
....:         return EvenPolynomialFunctor(), self.base()[self.variable_name()]
....:     def _coerce_map_from_(self, R):
....:         return self.base().has_coerce_map_from(R)
sage: class EvenPolynomialFunctor(ConstructionFunctor):
....:     rank = 10
....:     coercion_reversed = True
....:     def __init__(self):
....:         ConstructionFunctor.__init__(self, Rings(), Rings())
....:     def _apply_functor(self, R):
....:         return EvenPolynomialRing(R.base(), R.variable_name())
sage: pushout(EvenPolynomialRing(QQ, 'x'), ZZ)
Even Power Univariate Polynomial Ring in x over Rational Field
sage: pushout(EvenPolynomialRing(QQ, 'x'), QQ)
Even Power Univariate Polynomial Ring in x over Rational Field
sage: pushout(EvenPolynomialRing(QQ, 'x'), RR)                                  # needs sage.rings.real_mpfr
Even Power Univariate Polynomial Ring in x over Real Field with 53 bits of precision

sage: pushout(EvenPolynomialRing(QQ, 'x'), ZZ['x'])
Univariate Polynomial Ring in x over Rational Field
sage: pushout(EvenPolynomialRing(QQ, 'x'), QQ['x'])
Univariate Polynomial Ring in x over Rational Field
sage: pushout(EvenPolynomialRing(QQ, 'x'), RR['x'])                             # needs sage.rings.real_mpfr
Univariate Polynomial Ring in x over Real Field with 53 bits of precision

sage: pushout(EvenPolynomialRing(QQ, 'x'), EvenPolynomialRing(QQ, 'x'))
Even Power Univariate Polynomial Ring in x over Rational Field
sage: pushout(EvenPolynomialRing(QQ, 'x'), EvenPolynomialRing(RR, 'x'))         # needs sage.rings.real_mpfr
Even Power Univariate Polynomial Ring in x over Real Field with 53 bits of precision

sage: pushout(EvenPolynomialRing(QQ, 'x')^2, RR^2)                              # needs sage.modules sage.rings.real_mpfr
Ambient free module of rank 2
 over the principal ideal domain Even Power Univariate Polynomial Ring in x
  over Real Field with 53 bits of precision
sage: pushout(EvenPolynomialRing(QQ, 'x')^2, RR['x']^2)                         # needs sage.modules sage.rings.real_mpfr
Ambient free module of rank 2
 over the principal ideal domain Univariate Polynomial Ring in x
 over Real Field with 53 bits of precision

>>> from sage.all import *
>>> from sage.categories.pushout import ConstructionFunctor
>>> class EvenPolynomialRing(type(QQ['x'])):
...     def __init__(self, base, var):
...         super().__init__(base, var)
...         self.register_embedding(base[var])
...     def __repr__(self):
...         return "Even Power " + super().__repr__()
...     def construction(self):
...         return EvenPolynomialFunctor(), self.base()[self.variable_name()]
...     def _coerce_map_from_(self, R):
...         return self.base().has_coerce_map_from(R)
>>> class EvenPolynomialFunctor(ConstructionFunctor):
...     rank = Integer(10)
...     coercion_reversed = True
...     def __init__(self):
...         ConstructionFunctor.__init__(self, Rings(), Rings())
...     def _apply_functor(self, R):
...         return EvenPolynomialRing(R.base(), R.variable_name())
>>> pushout(EvenPolynomialRing(QQ, 'x'), ZZ)
Even Power Univariate Polynomial Ring in x over Rational Field
>>> pushout(EvenPolynomialRing(QQ, 'x'), QQ)
Even Power Univariate Polynomial Ring in x over Rational Field
>>> pushout(EvenPolynomialRing(QQ, 'x'), RR)                                  # needs sage.rings.real_mpfr
Even Power Univariate Polynomial Ring in x over Real Field with 53 bits of precision

>>> pushout(EvenPolynomialRing(QQ, 'x'), ZZ['x'])
Univariate Polynomial Ring in x over Rational Field
>>> pushout(EvenPolynomialRing(QQ, 'x'), QQ['x'])
Univariate Polynomial Ring in x over Rational Field
>>> pushout(EvenPolynomialRing(QQ, 'x'), RR['x'])                             # needs sage.rings.real_mpfr
Univariate Polynomial Ring in x over Real Field with 53 bits of precision

>>> pushout(EvenPolynomialRing(QQ, 'x'), EvenPolynomialRing(QQ, 'x'))
Even Power Univariate Polynomial Ring in x over Rational Field
>>> pushout(EvenPolynomialRing(QQ, 'x'), EvenPolynomialRing(RR, 'x'))         # needs sage.rings.real_mpfr
Even Power Univariate Polynomial Ring in x over Real Field with 53 bits of precision

>>> pushout(EvenPolynomialRing(QQ, 'x')**Integer(2), RR**Integer(2))                              # needs sage.modules sage.rings.real_mpfr
Ambient free module of rank 2
 over the principal ideal domain Even Power Univariate Polynomial Ring in x
  over Real Field with 53 bits of precision
>>> pushout(EvenPolynomialRing(QQ, 'x')**Integer(2), RR['x']**Integer(2))                         # needs sage.modules sage.rings.real_mpfr
Ambient free module of rank 2
 over the principal ideal domain Univariate Polynomial Ring in x
 over Real Field with 53 bits of precision

Some more tests related to univariate/multivariate constructions. We consider a generalization of polynomial rings, where in addition to the coefficient ring \(C\) we also specify an additive monoid \(E\) for the exponents of the indeterminate. In particular, the elements of such a parent are given by

\[\sum_{i=0}^I c_i X^{e_i}\]

with \(c_i \in C\) and \(e_i \in E\). We define

sage: class GPolynomialRing(Parent):
....:     def __init__(self, coefficients, var, exponents):
....:         self.coefficients = coefficients
....:         self.var = var
....:         self.exponents = exponents
....:         super().__init__(category=Rings())
....:     def _repr_(self):
....:         return 'Generalized Polynomial Ring in %s^(%s) over %s' % (
....:                self.var, self.exponents, self.coefficients)
....:     def construction(self):
....:         return GPolynomialFunctor(self.var, self.exponents), self.coefficients
....:     def _coerce_map_from_(self, R):
....:         return self.coefficients.has_coerce_map_from(R)

>>> from sage.all import *
>>> class GPolynomialRing(Parent):
...     def __init__(self, coefficients, var, exponents):
...         self.coefficients = coefficients
...         self.var = var
...         self.exponents = exponents
...         super().__init__(category=Rings())
...     def _repr_(self):
...         return 'Generalized Polynomial Ring in %s^(%s) over %s' % (
...                self.var, self.exponents, self.coefficients)
...     def construction(self):
...         return GPolynomialFunctor(self.var, self.exponents), self.coefficients
...     def _coerce_map_from_(self, R):
...         return self.coefficients.has_coerce_map_from(R)

and

sage: class GPolynomialFunctor(ConstructionFunctor):
....:     rank = 10
....:     def __init__(self, var, exponents):
....:         self.var = var
....:         self.exponents = exponents
....:         ConstructionFunctor.__init__(self, Rings(), Rings())
....:     def _repr_(self):
....:         return 'GPoly[%s^(%s)]' % (self.var, self.exponents)
....:     def _apply_functor(self, coefficients):
....:         return GPolynomialRing(coefficients, self.var, self.exponents)
....:     def merge(self, other):
....:         if isinstance(other, GPolynomialFunctor) and self.var == other.var:
....:             exponents = pushout(self.exponents, other.exponents)
....:             return GPolynomialFunctor(self.var, exponents)

>>> from sage.all import *
>>> class GPolynomialFunctor(ConstructionFunctor):
...     rank = Integer(10)
...     def __init__(self, var, exponents):
...         self.var = var
...         self.exponents = exponents
...         ConstructionFunctor.__init__(self, Rings(), Rings())
...     def _repr_(self):
...         return 'GPoly[%s^(%s)]' % (self.var, self.exponents)
...     def _apply_functor(self, coefficients):
...         return GPolynomialRing(coefficients, self.var, self.exponents)
...     def merge(self, other):
...         if isinstance(other, GPolynomialFunctor) and self.var == other.var:
...             exponents = pushout(self.exponents, other.exponents)
...             return GPolynomialFunctor(self.var, exponents)

We can construct a parent now in two different ways:

sage: GPolynomialRing(QQ, 'X', ZZ)
Generalized Polynomial Ring in X^(Integer Ring) over Rational Field
sage: GP_ZZ = GPolynomialFunctor('X', ZZ); GP_ZZ
GPoly[X^(Integer Ring)]
sage: GP_ZZ(QQ)
Generalized Polynomial Ring in X^(Integer Ring) over Rational Field

>>> from sage.all import *
>>> GPolynomialRing(QQ, 'X', ZZ)
Generalized Polynomial Ring in X^(Integer Ring) over Rational Field
>>> GP_ZZ = GPolynomialFunctor('X', ZZ); GP_ZZ
GPoly[X^(Integer Ring)]
>>> GP_ZZ(QQ)
Generalized Polynomial Ring in X^(Integer Ring) over Rational Field

Since the construction

sage: GP_ZZ(QQ).construction()
(GPoly[X^(Integer Ring)], Rational Field)

>>> from sage.all import *
>>> GP_ZZ(QQ).construction()
(GPoly[X^(Integer Ring)], Rational Field)

uses the coefficient ring, we have the usual coercion with respect to this parameter:

sage: pushout(GP_ZZ(ZZ), GP_ZZ(QQ))
Generalized Polynomial Ring in X^(Integer Ring) over Rational Field
sage: pushout(GP_ZZ(ZZ['t']), GP_ZZ(QQ))
Generalized Polynomial Ring in X^(Integer Ring)
  over Univariate Polynomial Ring in t over Rational Field
sage: pushout(GP_ZZ(ZZ['a,b']), GP_ZZ(ZZ['b,c']))
Generalized Polynomial Ring in X^(Integer Ring)
  over Multivariate Polynomial Ring in a, b, c over Integer Ring
sage: pushout(GP_ZZ(ZZ['a,b']), GP_ZZ(QQ['b,c']))
Generalized Polynomial Ring in X^(Integer Ring)
  over Multivariate Polynomial Ring in a, b, c over Rational Field
sage: pushout(GP_ZZ(ZZ['a,b']), GP_ZZ(ZZ['c,d']))
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension', ...)

>>> from sage.all import *
>>> pushout(GP_ZZ(ZZ), GP_ZZ(QQ))
Generalized Polynomial Ring in X^(Integer Ring) over Rational Field
>>> pushout(GP_ZZ(ZZ['t']), GP_ZZ(QQ))
Generalized Polynomial Ring in X^(Integer Ring)
  over Univariate Polynomial Ring in t over Rational Field
>>> pushout(GP_ZZ(ZZ['a,b']), GP_ZZ(ZZ['b,c']))
Generalized Polynomial Ring in X^(Integer Ring)
  over Multivariate Polynomial Ring in a, b, c over Integer Ring
>>> pushout(GP_ZZ(ZZ['a,b']), GP_ZZ(QQ['b,c']))
Generalized Polynomial Ring in X^(Integer Ring)
  over Multivariate Polynomial Ring in a, b, c over Rational Field
>>> pushout(GP_ZZ(ZZ['a,b']), GP_ZZ(ZZ['c,d']))
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension', ...)

sage: GP_QQ = GPolynomialFunctor('X', QQ)
sage: pushout(GP_ZZ(ZZ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Rational Field) over Integer Ring
sage: pushout(GP_QQ(ZZ), GP_ZZ(ZZ))
Generalized Polynomial Ring in X^(Rational Field) over Integer Ring

>>> from sage.all import *
>>> GP_QQ = GPolynomialFunctor('X', QQ)
>>> pushout(GP_ZZ(ZZ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Rational Field) over Integer Ring
>>> pushout(GP_QQ(ZZ), GP_ZZ(ZZ))
Generalized Polynomial Ring in X^(Rational Field) over Integer Ring

sage: GP_ZZt = GPolynomialFunctor('X', ZZ['t'])
sage: pushout(GP_ZZt(ZZ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Univariate Polynomial Ring in t
  over Rational Field) over Integer Ring

>>> from sage.all import *
>>> GP_ZZt = GPolynomialFunctor('X', ZZ['t'])
>>> pushout(GP_ZZt(ZZ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Univariate Polynomial Ring in t
  over Rational Field) over Integer Ring

sage: pushout(GP_ZZ(ZZ), GP_QQ(QQ))
Generalized Polynomial Ring in X^(Rational Field) over Rational Field
sage: pushout(GP_ZZ(QQ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Rational Field) over Rational Field
sage: pushout(GP_ZZt(QQ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Univariate Polynomial Ring in t
  over Rational Field) over Rational Field
sage: pushout(GP_ZZt(ZZ), GP_QQ(QQ))
Generalized Polynomial Ring in X^(Univariate Polynomial Ring in t
  over Rational Field) over Rational Field
sage: pushout(GP_ZZt(ZZ['a,b']), GP_QQ(ZZ['c,d']))
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension', ...)
sage: pushout(GP_ZZt(ZZ['a,b']), GP_QQ(ZZ['b,c']))
Generalized Polynomial Ring
 in X^(Univariate Polynomial Ring in t over Rational Field)
  over Multivariate Polynomial Ring in a, b, c over Integer Ring

>>> from sage.all import *
>>> pushout(GP_ZZ(ZZ), GP_QQ(QQ))
Generalized Polynomial Ring in X^(Rational Field) over Rational Field
>>> pushout(GP_ZZ(QQ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Rational Field) over Rational Field
>>> pushout(GP_ZZt(QQ), GP_QQ(ZZ))
Generalized Polynomial Ring in X^(Univariate Polynomial Ring in t
  over Rational Field) over Rational Field
>>> pushout(GP_ZZt(ZZ), GP_QQ(QQ))
Generalized Polynomial Ring in X^(Univariate Polynomial Ring in t
  over Rational Field) over Rational Field
>>> pushout(GP_ZZt(ZZ['a,b']), GP_QQ(ZZ['c,d']))
Traceback (most recent call last):
...
CoercionException: ('Ambiguous Base Extension', ...)
>>> pushout(GP_ZZt(ZZ['a,b']), GP_QQ(ZZ['b,c']))
Generalized Polynomial Ring
 in X^(Univariate Polynomial Ring in t over Rational Field)
  over Multivariate Polynomial Ring in a, b, c over Integer Ring

Some tests with Cartesian products:

sage: from sage.sets.cartesian_product import CartesianProduct
sage: A = CartesianProduct((ZZ['x'], QQ['y'], QQ['z']),
....:                      Sets().CartesianProducts())
sage: B = CartesianProduct((ZZ['x'], ZZ['y'], ZZ['t']['z']),
....:                      Sets().CartesianProducts())
sage: A.construction()
(The cartesian_product functorial construction,
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Rational Field,
  Univariate Polynomial Ring in z over Rational Field))
sage: pushout(A, B)
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Rational Field,
  Univariate Polynomial Ring in z over
   Univariate Polynomial Ring in t over Rational Field)
sage: pushout(ZZ, cartesian_product([ZZ, QQ]))
Traceback (most recent call last):
...
CoercionException: 'NoneType' object is not iterable

>>> from sage.all import *
>>> from sage.sets.cartesian_product import CartesianProduct
>>> A = CartesianProduct((ZZ['x'], QQ['y'], QQ['z']),
...                      Sets().CartesianProducts())
>>> B = CartesianProduct((ZZ['x'], ZZ['y'], ZZ['t']['z']),
...                      Sets().CartesianProducts())
>>> A.construction()
(The cartesian_product functorial construction,
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Rational Field,
  Univariate Polynomial Ring in z over Rational Field))
>>> pushout(A, B)
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Rational Field,
  Univariate Polynomial Ring in z over
   Univariate Polynomial Ring in t over Rational Field)
>>> pushout(ZZ, cartesian_product([ZZ, QQ]))
Traceback (most recent call last):
...
CoercionException: 'NoneType' object is not iterable

sage: from sage.categories.pushout import PolynomialFunctor
sage: from sage.sets.cartesian_product import CartesianProduct
sage: class CartesianProductPoly(CartesianProduct):
....:     def __init__(self, polynomial_rings):
....:         sort = sorted(polynomial_rings,
....:                       key=lambda P: P.variable_name())
....:         super().__init__(sort, Sets().CartesianProducts())
....:     def vars(self):
....:         return tuple(P.variable_name()
....:                      for P in self.cartesian_factors())
....:     def _pushout_(self, other):
....:         if isinstance(other, CartesianProductPoly):
....:             s_vars = self.vars()
....:             o_vars = other.vars()
....:             if s_vars == o_vars:
....:                 return
....:             return pushout(CartesianProductPoly(
....:                     self.cartesian_factors() +
....:                     tuple(f for f in other.cartesian_factors()
....:                           if f.variable_name() not in s_vars)),
....:                 CartesianProductPoly(
....:                     other.cartesian_factors() +
....:                     tuple(f for f in self.cartesian_factors()
....:                           if f.variable_name() not in o_vars)))
....:         C = other.construction()
....:         if C is None:
....:             return
....:         elif isinstance(C[0], PolynomialFunctor):
....:             return pushout(self, CartesianProductPoly((other,)))

>>> from sage.all import *
>>> from sage.categories.pushout import PolynomialFunctor
>>> from sage.sets.cartesian_product import CartesianProduct
>>> class CartesianProductPoly(CartesianProduct):
...     def __init__(self, polynomial_rings):
...         sort = sorted(polynomial_rings,
...                       key=lambda P: P.variable_name())
...         super().__init__(sort, Sets().CartesianProducts())
...     def vars(self):
...         return tuple(P.variable_name()
...                      for P in self.cartesian_factors())
...     def _pushout_(self, other):
...         if isinstance(other, CartesianProductPoly):
...             s_vars = self.vars()
...             o_vars = other.vars()
...             if s_vars == o_vars:
...                 return
...             return pushout(CartesianProductPoly(
...                     self.cartesian_factors() +
...                     tuple(f for f in other.cartesian_factors()
...                           if f.variable_name() not in s_vars)),
...                 CartesianProductPoly(
...                     other.cartesian_factors() +
...                     tuple(f for f in self.cartesian_factors()
...                           if f.variable_name() not in o_vars)))
...         C = other.construction()
...         if C is None:
...             return
...         elif isinstance(C[Integer(0)], PolynomialFunctor):
...             return pushout(self, CartesianProductPoly((other,)))

sage: pushout(CartesianProductPoly((ZZ['x'],)),
....:         CartesianProductPoly((ZZ['y'],)))
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Integer Ring)
sage: pushout(CartesianProductPoly((ZZ['x'], ZZ['y'])),
....:         CartesianProductPoly((ZZ['x'], ZZ['z'])))
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Integer Ring,
  Univariate Polynomial Ring in z over Integer Ring)
sage: pushout(CartesianProductPoly((QQ['a,b']['x'], QQ['y'])),                  # needs sage.symbolic
....:         CartesianProductPoly((ZZ['b,c']['x'], SR['z'])))
The Cartesian product of
 (Univariate Polynomial Ring in x over
    Multivariate Polynomial Ring in a, b, c over Rational Field,
  Univariate Polynomial Ring in y over Rational Field,
  Univariate Polynomial Ring in z over Symbolic Ring)

>>> from sage.all import *
>>> pushout(CartesianProductPoly((ZZ['x'],)),
...         CartesianProductPoly((ZZ['y'],)))
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Integer Ring)
>>> pushout(CartesianProductPoly((ZZ['x'], ZZ['y'])),
...         CartesianProductPoly((ZZ['x'], ZZ['z'])))
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Integer Ring,
  Univariate Polynomial Ring in z over Integer Ring)
>>> pushout(CartesianProductPoly((QQ['a,b']['x'], QQ['y'])),                  # needs sage.symbolic
...         CartesianProductPoly((ZZ['b,c']['x'], SR['z'])))
The Cartesian product of
 (Univariate Polynomial Ring in x over
    Multivariate Polynomial Ring in a, b, c over Rational Field,
  Univariate Polynomial Ring in y over Rational Field,
  Univariate Polynomial Ring in z over Symbolic Ring)

sage: pushout(CartesianProductPoly((ZZ['x'],)), ZZ['y'])
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Integer Ring)
sage: pushout(QQ['b,c']['y'], CartesianProductPoly((ZZ['a,b']['x'],)))
The Cartesian product of
 (Univariate Polynomial Ring in x over
    Multivariate Polynomial Ring in a, b over Integer Ring,
  Univariate Polynomial Ring in y over
    Multivariate Polynomial Ring in b, c over Rational Field)

>>> from sage.all import *
>>> pushout(CartesianProductPoly((ZZ['x'],)), ZZ['y'])
The Cartesian product of
 (Univariate Polynomial Ring in x over Integer Ring,
  Univariate Polynomial Ring in y over Integer Ring)
>>> pushout(QQ['b,c']['y'], CartesianProductPoly((ZZ['a,b']['x'],)))
The Cartesian product of
 (Univariate Polynomial Ring in x over
    Multivariate Polynomial Ring in a, b over Integer Ring,
  Univariate Polynomial Ring in y over
    Multivariate Polynomial Ring in b, c over Rational Field)

sage: pushout(CartesianProductPoly((ZZ['x'],)), ZZ)
Traceback (most recent call last):
...
CoercionException: No common base ("join") found for
The cartesian_product functorial construction(...) and None(Integer Ring):
(Multivariate) functors are incompatible.

>>> from sage.all import *
>>> pushout(CartesianProductPoly((ZZ['x'],)), ZZ)
Traceback (most recent call last):
...
CoercionException: No common base ("join") found for
The cartesian_product functorial construction(...) and None(Integer Ring):
(Multivariate) functors are incompatible.

AUTHORS:

  • Robert Bradshaw

  • Peter Bruin

  • Simon King

  • Daniel Krenn

  • David Roe

sage.categories.pushout.pushout_lattice(R, S)[source]#

Given a pair of objects \(R\) and \(S\), try to construct a reasonable object \(Y\) and return maps such that canonically \(R \leftarrow Y \rightarrow S\).

ALGORITHM:

This is based on the model that arose from much discussion at Sage Days 4. Going up the tower of constructions of \(R\) and \(S\) (e.g. the reals come from the rationals come from the integers), try to find a common parent, and then try to fill in a lattice with these two towers as sides with the top as the common ancestor and the bottom will be the desired ring.

See the code for a specific worked-out example.

EXAMPLES:

sage: from sage.categories.pushout import pushout_lattice
sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x']))                              # needs sage.rings.padics
sage: A.codomain()                                                              # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20
sage: A.codomain() is B.codomain()                                              # needs sage.rings.padics
True
sage: A, B = pushout_lattice(ZZ, MatrixSpace(ZZ[['x']], 3, 3))                  # needs sage.modules
sage: B                                                                         # needs sage.modules
Identity endomorphism of Full MatrixSpace of 3 by 3 dense matrices
 over Power Series Ring in x over Integer Ring

>>> from sage.all import *
>>> from sage.categories.pushout import pushout_lattice
>>> A, B = pushout_lattice(Qp(Integer(7)), Frac(ZZ['x']))                              # needs sage.rings.padics
>>> A.codomain()                                                              # needs sage.rings.padics
Fraction Field of Univariate Polynomial Ring in x
 over 7-adic Field with capped relative precision 20
>>> A.codomain() is B.codomain()                                              # needs sage.rings.padics
True
>>> A, B = pushout_lattice(ZZ, MatrixSpace(ZZ[['x']], Integer(3), Integer(3)))                  # needs sage.modules
>>> B                                                                         # needs sage.modules
Identity endomorphism of Full MatrixSpace of 3 by 3 dense matrices
 over Power Series Ring in x over Integer Ring

AUTHOR:

  • Robert Bradshaw

sage.categories.pushout.type_to_parent(P)[source]#

An auxiliary function that is used in pushout().

INPUT:

A type

OUTPUT:

A Sage parent structure corresponding to the given type