Coerce actions

class sage.structure.coerce_actions.ActOnAction[source]

Bases: GenericAction

Class for actions defined via the _act_on_ method.

class sage.structure.coerce_actions.ActedUponAction[source]

Bases: GenericAction

Class for actions defined via the _acted_upon_ method.

class sage.structure.coerce_actions.GenericAction[source]

Bases: Action

codomain()[source]

Return the “codomain” of this action, i.e. the Parent in which the result elements live. Typically, this should be the same as the acted upon set.

EXAMPLES:

Note that coerce actions should only be used inside of the coercion model. For this test, we need to strongly reference the domains, for otherwise they could be garbage collected, giving rise to random errors (see Issue #18157).

sage: M = MatrixSpace(ZZ, 2)                                                # needs sage.modules
sage: A = sage.structure.coerce_actions.ActedUponAction(M, Cusps, True)     # needs sage.modular sage.modules
sage: A.codomain()                                                          # needs sage.modular sage.modules
Set P^1(QQ) of all cusps

sage: # needs sage.groups
sage: S3 = SymmetricGroup(3)
sage: QQxyz = QQ['x,y,z']
sage: A = sage.structure.coerce_actions.ActOnAction(S3, QQxyz, False)
sage: A.codomain()
Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> M = MatrixSpace(ZZ, Integer(2))                                                # needs sage.modules
>>> A = sage.structure.coerce_actions.ActedUponAction(M, Cusps, True)     # needs sage.modular sage.modules
>>> A.codomain()                                                          # needs sage.modular sage.modules
Set P^1(QQ) of all cusps

>>> # needs sage.groups
>>> S3 = SymmetricGroup(Integer(3))
>>> QQxyz = QQ['x,y,z']
>>> A = sage.structure.coerce_actions.ActOnAction(S3, QQxyz, False)
>>> A.codomain()
Multivariate Polynomial Ring in x, y, z over Rational Field

M = MatrixSpace(ZZ, 2)                                                # needs sage.modules
A = sage.structure.coerce_actions.ActedUponAction(M, Cusps, True)     # needs sage.modular sage.modules
A.codomain()                                                          # needs sage.modular sage.modules
# needs sage.groups
S3 = SymmetricGroup(3)
QQxyz = QQ['x,y,z']
A = sage.structure.coerce_actions.ActOnAction(S3, QQxyz, False)
A.codomain()
class sage.structure.coerce_actions.IntegerAction[source]

Bases: Action

Abstract base class representing some action by integers on something. Here, “integer” is defined loosely in the “duck typing” sense.

INPUT:

  • Z – a type or parent representing integers

For the other arguments, see Action.

Note

This class is used internally in Sage’s coercion model. Outside of the coercion model, special precautions are needed to prevent domains of the action from being garbage collected.

class sage.structure.coerce_actions.IntegerMulAction[source]

Bases: IntegerAction

Implement the action \(n \cdot a = a + a + ... + a\) via repeated doubling.

Both addition and negation must be defined on the set \(M\).

INPUT:

  • Z – a type or parent representing integers

  • M – a ZZ-module

  • m – (optional) an element of M

EXAMPLES:

sage: from sage.structure.coerce_actions import IntegerMulAction
sage: R.<x> = QQ['x']
sage: act = IntegerMulAction(ZZ, R)
sage: act(5, x)
5*x
sage: act(0, x)
0
sage: act(-3, x-1)
-3*x + 3

>>> from sage.all import *
>>> from sage.structure.coerce_actions import IntegerMulAction
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> act = IntegerMulAction(ZZ, R)
>>> act(Integer(5), x)
5*x
>>> act(Integer(0), x)
0
>>> act(-Integer(3), x-Integer(1))
-3*x + 3

from sage.structure.coerce_actions import IntegerMulAction
R.<x> = QQ['x']
act = IntegerMulAction(ZZ, R)
act(5, x)
act(0, x)
act(-3, x-1)
class sage.structure.coerce_actions.IntegerPowAction[source]

Bases: IntegerAction

The right action a ^ n = a * a * ... * a where \(n\) is an integer.

The action is implemented using the _pow_int method on elements.

INPUT:

  • Z – a type or parent representing integers

  • M – a parent whose elements implement _pow_int

  • m – (optional) an element of M

EXAMPLES:

sage: from sage.structure.coerce_actions import IntegerPowAction
sage: R.<x> = LaurentSeriesRing(QQ)
sage: act = IntegerPowAction(ZZ, R)
sage: act(x, 5)
x^5
sage: act(x, -2)
x^-2
sage: act(x, int(5))
x^5

>>> from sage.all import *
>>> from sage.structure.coerce_actions import IntegerPowAction
>>> R = LaurentSeriesRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> act = IntegerPowAction(ZZ, R)
>>> act(x, Integer(5))
x^5
>>> act(x, -Integer(2))
x^-2
>>> act(x, int(Integer(5)))
x^5

from sage.structure.coerce_actions import IntegerPowAction
R.<x> = LaurentSeriesRing(QQ)
act = IntegerPowAction(ZZ, R)
act(x, 5)
act(x, -2)
act(x, int(5))
class sage.structure.coerce_actions.LeftModuleAction[source]

Bases: ModuleAction

class sage.structure.coerce_actions.ModuleAction[source]

Bases: Action

Module action.

See also

This is an abstract class, one must actually instantiate a LeftModuleAction or a RightModuleAction.

INPUT:

  • G – the actor, an instance of Parent

  • S – the object that is acted upon

  • g – (optional) an element of G

  • a – (optional) an element of S

  • check – if True (default), then there will be no consistency tests performed on sample elements

NOTE:

By default, the sample elements of S and G are obtained from an_element(), which relies on the implementation of an _an_element_() method. This is not always available. But usually, the action is only needed when one already has two elements. Hence, by Issue #14249, the coercion model will pass these two elements to the ModuleAction constructor.

The actual action is implemented by the _rmul_ or _lmul_ function on its elements. We must, however, be very particular about what we feed into these functions, because they operate under the assumption that the inputs lie exactly in the base ring and may segfault otherwise. Thus we handle all possible base extensions manually here.

codomain()[source]

The codomain of self, which may or may not be equal to the domain.

EXAMPLES:

Note that coerce actions should only be used inside of the coercion model. For this test, we need to strongly reference the domains, for otherwise they could be garbage collected, giving rise to random errors (see Issue #18157).

sage: from sage.structure.coerce_actions import LeftModuleAction
sage: ZZxyz = ZZ['x,y,z']
sage: A = LeftModuleAction(QQ, ZZxyz)
sage: A.codomain()
Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> from sage.structure.coerce_actions import LeftModuleAction
>>> ZZxyz = ZZ['x,y,z']
>>> A = LeftModuleAction(QQ, ZZxyz)
>>> A.codomain()
Multivariate Polynomial Ring in x, y, z over Rational Field

from sage.structure.coerce_actions import LeftModuleAction
ZZxyz = ZZ['x,y,z']
A = LeftModuleAction(QQ, ZZxyz)
A.codomain()
domain()[source]

The domain of self, which is the module that is being acted on.

EXAMPLES:

Note that coerce actions should only be used inside of the coercion model. For this test, we need to strongly reference the domains, for otherwise they could be garbage collected, giving rise to random errors (see Issue #18157).

sage: from sage.structure.coerce_actions import LeftModuleAction
sage: ZZxyz = ZZ['x,y,z']
sage: A = LeftModuleAction(QQ, ZZxyz)
sage: A.domain()
Multivariate Polynomial Ring in x, y, z over Integer Ring

>>> from sage.all import *
>>> from sage.structure.coerce_actions import LeftModuleAction
>>> ZZxyz = ZZ['x,y,z']
>>> A = LeftModuleAction(QQ, ZZxyz)
>>> A.domain()
Multivariate Polynomial Ring in x, y, z over Integer Ring

from sage.structure.coerce_actions import LeftModuleAction
ZZxyz = ZZ['x,y,z']
A = LeftModuleAction(QQ, ZZxyz)
A.domain()
class sage.structure.coerce_actions.PyScalarAction[source]

Bases: Action

class sage.structure.coerce_actions.RightModuleAction[source]

Bases: ModuleAction

sage.structure.coerce_actions.detect_element_action(X, Y, X_on_left, X_el=None, Y_el=None)[source]

Return an action of X on Y as defined by elements of X, if any.

EXAMPLES:

Note that coerce actions should only be used inside of the coercion model. For this test, we need to strongly reference the domains, for otherwise they could be garbage collected, giving rise to random errors (see Issue #18157).

sage: from sage.structure.coerce_actions import detect_element_action
sage: ZZx = ZZ['x']
sage: M = MatrixSpace(ZZ, 2)                                                    # needs sage.modules
sage: detect_element_action(ZZx, ZZ, False)
Left scalar multiplication by Integer Ring
 on Univariate Polynomial Ring in x over Integer Ring
sage: detect_element_action(ZZx, QQ, True)
Right scalar multiplication by Rational Field
 on Univariate Polynomial Ring in x over Integer Ring
sage: detect_element_action(Cusps, M, False)                                    # needs sage.modular sage.modules
Left action by Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
 on Set P^1(QQ) of all cusps
sage: detect_element_action(Cusps, M, True),                                    # needs sage.modular sage.modules
(None,)
sage: detect_element_action(ZZ, QQ, True),
(None,)

>>> from sage.all import *
>>> from sage.structure.coerce_actions import detect_element_action
>>> ZZx = ZZ['x']
>>> M = MatrixSpace(ZZ, Integer(2))                                                    # needs sage.modules
>>> detect_element_action(ZZx, ZZ, False)
Left scalar multiplication by Integer Ring
 on Univariate Polynomial Ring in x over Integer Ring
>>> detect_element_action(ZZx, QQ, True)
Right scalar multiplication by Rational Field
 on Univariate Polynomial Ring in x over Integer Ring
>>> detect_element_action(Cusps, M, False)                                    # needs sage.modular sage.modules
Left action by Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
 on Set P^1(QQ) of all cusps
>>> detect_element_action(Cusps, M, True),                                    # needs sage.modular sage.modules
(None,)
>>> detect_element_action(ZZ, QQ, True),
(None,)

from sage.structure.coerce_actions import detect_element_action
ZZx = ZZ['x']
M = MatrixSpace(ZZ, 2)                                                    # needs sage.modules
detect_element_action(ZZx, ZZ, False)
detect_element_action(ZZx, QQ, True)
detect_element_action(Cusps, M, False)                                    # needs sage.modular sage.modules
detect_element_action(Cusps, M, True),                                    # needs sage.modular sage.modules
detect_element_action(ZZ, QQ, True),