Subrings of the Symbolic Ring

Subrings of the symbolic ring can be created via the subring() method of SR. This will call SymbolicSubring of this module.

The following kinds of subrings are supported:

  • A symbolic subring of expressions, whose variables are contained in a given set of symbolic variables (see SymbolicSubringAcceptingVars). E.g.

    sage: SR.subring(accepting_variables=('a', 'b'))
Symbolic Subring accepting the variables a, b

    >>> from sage.all import *
>>> SR.subring(accepting_variables=('a', 'b'))
Symbolic Subring accepting the variables a, b

    SR.subring(accepting_variables=('a', 'b'))

  • A symbolic subring of expressions, whose variables are disjoint to a given set of symbolic variables (see SymbolicSubringRejectingVars). E.g.

    sage: SR.subring(rejecting_variables=('r', 's'))
Symbolic Subring rejecting the variables r, s

    >>> from sage.all import *
>>> SR.subring(rejecting_variables=('r', 's'))
Symbolic Subring rejecting the variables r, s

    SR.subring(rejecting_variables=('r', 's'))

  • The subring of symbolic constants (see SymbolicConstantsSubring). E.g.

    sage: SR.subring(no_variables=True)
Symbolic Constants Subring

    >>> from sage.all import *
>>> SR.subring(no_variables=True)
Symbolic Constants Subring

    SR.subring(no_variables=True)

AUTHORS:

  • Daniel Krenn (2015)

Classes and Methods

class sage.symbolic.subring.GenericSymbolicSubring(vars)[source]

Bases: SymbolicRing

An abstract base class for a symbolic subring.

INPUT:

  • vars – tuple of symbolic variables

has_valid_variable(variable)[source]

Return whether the given variable is valid in this subring.

INPUT:

  • variable – a symbolic variable

OUTPUT: boolean

EXAMPLES:

sage: from sage.symbolic.subring import GenericSymbolicSubring
sage: GenericSymbolicSubring(vars=tuple()).has_valid_variable(x)
Traceback (most recent call last):
...
NotImplementedError: Not implemented in this abstract base class

>>> from sage.all import *
>>> from sage.symbolic.subring import GenericSymbolicSubring
>>> GenericSymbolicSubring(vars=tuple()).has_valid_variable(x)
Traceback (most recent call last):
...
NotImplementedError: Not implemented in this abstract base class

from sage.symbolic.subring import GenericSymbolicSubring
GenericSymbolicSubring(vars=tuple()).has_valid_variable(x)
class sage.symbolic.subring.GenericSymbolicSubringFunctor(vars)[source]

Bases: ConstructionFunctor

A base class for the functors constructing symbolic subrings.

INPUT:

  • vars – tuple, set, or other iterable of symbolic variables

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: SymbolicSubring(no_variables=True).construction()[0]  # indirect doctest
Subring<accepting no variable>

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> SymbolicSubring(no_variables=True).construction()[Integer(0)]  # indirect doctest
Subring<accepting no variable>

from sage.symbolic.subring import SymbolicSubring
SymbolicSubring(no_variables=True).construction()[0]  # indirect doctest

See also

sage.categories.pushout.ConstructionFunctor.

coercion_reversed = True
merge(other)[source]

Merge this functor with other if possible.

INPUT:

  • other – a functor

OUTPUT: a functor or None

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: F = SymbolicSubring(accepting_variables=('a',)).construction()[0]
sage: F.merge(F) is F
True

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> F = SymbolicSubring(accepting_variables=('a',)).construction()[Integer(0)]
>>> F.merge(F) is F
True

from sage.symbolic.subring import SymbolicSubring
F = SymbolicSubring(accepting_variables=('a',)).construction()[0]
F.merge(F) is F
rank = 11
class sage.symbolic.subring.SymbolicConstantsSubring(vars)[source]

Bases: SymbolicSubringAcceptingVars

The symbolic subring consisting of symbolic constants.

has_valid_variable(variable)[source]

Return whether the given variable is valid in this subring.

INPUT:

  • variable – a symbolic variable

OUTPUT: boolean

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: S = SymbolicSubring(no_variables=True)
sage: S.has_valid_variable('a')
False
sage: S.has_valid_variable('r')
False
sage: S.has_valid_variable('x')
False

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> S = SymbolicSubring(no_variables=True)
>>> S.has_valid_variable('a')
False
>>> S.has_valid_variable('r')
False
>>> S.has_valid_variable('x')
False

from sage.symbolic.subring import SymbolicSubring
S = SymbolicSubring(no_variables=True)
S.has_valid_variable('a')
S.has_valid_variable('r')
S.has_valid_variable('x')
class sage.symbolic.subring.SymbolicSubringAcceptingVars(vars)[source]

Bases: GenericSymbolicSubring

The symbolic subring consisting of symbolic expressions in the given variables.

construction()[source]

Return the functorial construction of this symbolic subring.

OUTPUT:

A tuple whose first entry is a construction functor and its second is the symbolic ring.

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: SymbolicSubring(accepting_variables=('a',)).construction()
(Subring<accepting a>, Symbolic Ring)

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> SymbolicSubring(accepting_variables=('a',)).construction()
(Subring<accepting a>, Symbolic Ring)

from sage.symbolic.subring import SymbolicSubring
SymbolicSubring(accepting_variables=('a',)).construction()
has_valid_variable(variable)[source]

Return whether the given variable is valid in this subring.

INPUT:

  • variable – a symbolic variable

OUTPUT: boolean

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: S = SymbolicSubring(accepting_variables=('a',))
sage: S.has_valid_variable('a')
True
sage: S.has_valid_variable('r')
False
sage: S.has_valid_variable('x')
False

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> S = SymbolicSubring(accepting_variables=('a',))
>>> S.has_valid_variable('a')
True
>>> S.has_valid_variable('r')
False
>>> S.has_valid_variable('x')
False

from sage.symbolic.subring import SymbolicSubring
S = SymbolicSubring(accepting_variables=('a',))
S.has_valid_variable('a')
S.has_valid_variable('r')
S.has_valid_variable('x')
class sage.symbolic.subring.SymbolicSubringAcceptingVarsFunctor(vars)[source]

Bases: GenericSymbolicSubringFunctor

merge(other)[source]

Merge this functor with other if possible.

INPUT:

  • other – a functor

OUTPUT: a functor or None

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: F = SymbolicSubring(accepting_variables=('a',)).construction()[0]
sage: G = SymbolicSubring(rejecting_variables=('r',)).construction()[0]
sage: F.merge(F) is F
True
sage: F.merge(G) is G
True

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> F = SymbolicSubring(accepting_variables=('a',)).construction()[Integer(0)]
>>> G = SymbolicSubring(rejecting_variables=('r',)).construction()[Integer(0)]
>>> F.merge(F) is F
True
>>> F.merge(G) is G
True

from sage.symbolic.subring import SymbolicSubring
F = SymbolicSubring(accepting_variables=('a',)).construction()[0]
G = SymbolicSubring(rejecting_variables=('r',)).construction()[0]
F.merge(F) is F
F.merge(G) is G
class sage.symbolic.subring.SymbolicSubringFactory[source]

Bases: UniqueFactory

A factory creating a symbolic subring.

INPUT:

Specify one of the following keywords to create a subring.

  • accepting_variables – (default: None) a tuple or other iterable of variables. If specified, then a symbolic subring of expressions in only these variables is created.

  • rejecting_variables – (default: None) a tuple or other iterable of variables. If specified, then a symbolic subring of expressions in variables distinct to these variables is created.

  • no_variables – boolean (default: False); if set, then a symbolic subring of constant expressions (i.e., expressions without a variable) is created.

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: V = var('a, b, c, r, s, t, x, y, z')

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> V = var('a, b, c, r, s, t, x, y, z')

from sage.symbolic.subring import SymbolicSubring
V = var('a, b, c, r, s, t, x, y, z')
sage: A = SymbolicSubring(accepting_variables=(a, b, c)); A
Symbolic Subring accepting the variables a, b, c
sage: tuple((v, v in A) for v in V)
((a, True), (b, True), (c, True),
 (r, False), (s, False), (t, False),
 (x, False), (y, False), (z, False))

>>> from sage.all import *
>>> A = SymbolicSubring(accepting_variables=(a, b, c)); A
Symbolic Subring accepting the variables a, b, c
>>> tuple((v, v in A) for v in V)
((a, True), (b, True), (c, True),
 (r, False), (s, False), (t, False),
 (x, False), (y, False), (z, False))

A = SymbolicSubring(accepting_variables=(a, b, c)); A
tuple((v, v in A) for v in V)
>>> from sage.all import *
>>> A = SymbolicSubring(accepting_variables=(a, b, c)); A
Symbolic Subring accepting the variables a, b, c
>>> tuple((v, v in A) for v in V)
((a, True), (b, True), (c, True),
 (r, False), (s, False), (t, False),
 (x, False), (y, False), (z, False))

A = SymbolicSubring(accepting_variables=(a, b, c)); A
tuple((v, v in A) for v in V)
sage: R = SymbolicSubring(rejecting_variables=(r, s, t)); R
Symbolic Subring rejecting the variables r, s, t
sage: tuple((v, v in R) for v in V)
((a, True), (b, True), (c, True),
 (r, False), (s, False), (t, False),
 (x, True), (y, True), (z, True))

>>> from sage.all import *
>>> R = SymbolicSubring(rejecting_variables=(r, s, t)); R
Symbolic Subring rejecting the variables r, s, t
>>> tuple((v, v in R) for v in V)
((a, True), (b, True), (c, True),
 (r, False), (s, False), (t, False),
 (x, True), (y, True), (z, True))

R = SymbolicSubring(rejecting_variables=(r, s, t)); R
tuple((v, v in R) for v in V)
>>> from sage.all import *
>>> R = SymbolicSubring(rejecting_variables=(r, s, t)); R
Symbolic Subring rejecting the variables r, s, t
>>> tuple((v, v in R) for v in V)
((a, True), (b, True), (c, True),
 (r, False), (s, False), (t, False),
 (x, True), (y, True), (z, True))

R = SymbolicSubring(rejecting_variables=(r, s, t)); R
tuple((v, v in R) for v in V)
sage: C = SymbolicSubring(no_variables=True); C
Symbolic Constants Subring
sage: tuple((v, v in C) for v in V)
((a, False), (b, False), (c, False),
 (r, False), (s, False), (t, False),
 (x, False), (y, False), (z, False))

>>> from sage.all import *
>>> C = SymbolicSubring(no_variables=True); C
Symbolic Constants Subring
>>> tuple((v, v in C) for v in V)
((a, False), (b, False), (c, False),
 (r, False), (s, False), (t, False),
 (x, False), (y, False), (z, False))

C = SymbolicSubring(no_variables=True); C
tuple((v, v in C) for v in V)
>>> from sage.all import *
>>> C = SymbolicSubring(no_variables=True); C
Symbolic Constants Subring
>>> tuple((v, v in C) for v in V)
((a, False), (b, False), (c, False),
 (r, False), (s, False), (t, False),
 (x, False), (y, False), (z, False))

C = SymbolicSubring(no_variables=True); C
tuple((v, v in C) for v in V)
create_key_and_extra_args(accepting_variables=None, rejecting_variables=None, no_variables=False, **kwds)[source]

Given the arguments and keyword, create a key that uniquely determines this object.

See SymbolicSubringFactory for details.

create_object(version, key, **kwds)[source]

Create an object from the given arguments.

See SymbolicSubringFactory for details.

class sage.symbolic.subring.SymbolicSubringRejectingVars(vars)[source]

Bases: GenericSymbolicSubring

The symbolic subring consisting of symbolic expressions whose variables are none of the given variables.

construction()[source]

Return the functorial construction of this symbolic subring.

OUTPUT:

A tuple whose first entry is a construction functor and its second is the symbolic ring.

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: SymbolicSubring(rejecting_variables=('r',)).construction()
(Subring<rejecting r>, Symbolic Ring)

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> SymbolicSubring(rejecting_variables=('r',)).construction()
(Subring<rejecting r>, Symbolic Ring)

from sage.symbolic.subring import SymbolicSubring
SymbolicSubring(rejecting_variables=('r',)).construction()
has_valid_variable(variable)[source]

Return whether the given variable is valid in this subring.

INPUT:

  • variable – a symbolic variable

OUTPUT: boolean

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: S = SymbolicSubring(rejecting_variables=('r',))
sage: S.has_valid_variable('a')
True
sage: S.has_valid_variable('r')
False
sage: S.has_valid_variable('x')
True

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> S = SymbolicSubring(rejecting_variables=('r',))
>>> S.has_valid_variable('a')
True
>>> S.has_valid_variable('r')
False
>>> S.has_valid_variable('x')
True

from sage.symbolic.subring import SymbolicSubring
S = SymbolicSubring(rejecting_variables=('r',))
S.has_valid_variable('a')
S.has_valid_variable('r')
S.has_valid_variable('x')
class sage.symbolic.subring.SymbolicSubringRejectingVarsFunctor(vars)[source]

Bases: GenericSymbolicSubringFunctor

merge(other)[source]

Merge this functor with other if possible.

INPUT:

  • other – a functor

OUTPUT: a functor or None

EXAMPLES:

sage: from sage.symbolic.subring import SymbolicSubring
sage: F = SymbolicSubring(accepting_variables=('a',)).construction()[0]
sage: G = SymbolicSubring(rejecting_variables=('r',)).construction()[0]
sage: G.merge(G) is G
True
sage: G.merge(F) is G
True

>>> from sage.all import *
>>> from sage.symbolic.subring import SymbolicSubring
>>> F = SymbolicSubring(accepting_variables=('a',)).construction()[Integer(0)]
>>> G = SymbolicSubring(rejecting_variables=('r',)).construction()[Integer(0)]
>>> G.merge(G) is G
True
>>> G.merge(F) is G
True

from sage.symbolic.subring import SymbolicSubring
F = SymbolicSubring(accepting_variables=('a',)).construction()[0]
G = SymbolicSubring(rejecting_variables=('r',)).construction()[0]
G.merge(G) is G
G.merge(F) is G