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
```
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
```
The subring of symbolic constants (see SymbolicConstantsSubring). E.g.
```
sage: SR.subring(no_variables=True)
Symbolic Constants Subring
```

AUTHORS:

Daniel Krenn (2015)

Classes and Methods¶

class sage.symbolic.subring.GenericSymbolicSubring(vars)¶

Bases: sage.symbolic.ring.SymbolicRing

An abstract base class for a symbolic subring.

INPUT:

vars – a tuple of symbolic variables.

has_valid_variable(variable)¶

Return whether the given variable is valid in this subring.

INPUT:

variable – a symbolic variable.

OUTPUT:

A 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

class sage.symbolic.subring.GenericSymbolicSubringFunctor(vars)¶

Bases: sage.categories.pushout.ConstructionFunctor

A base class for the functors constructing symbolic subrings.

INPUT:

vars – a 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>

merge(other)¶

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

class sage.symbolic.subring.SymbolicConstantsSubring(vars)¶

Bases: sage.symbolic.subring.SymbolicSubringAcceptingVars

The symbolic subring consisting of symbolic constants.

has_valid_variable(variable)¶

Return whether the given variable is valid in this subring.

INPUT:

variable – a symbolic variable.

OUTPUT:

A 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

class sage.symbolic.subring.SymbolicSubringAcceptingVars(vars)¶

Bases: sage.symbolic.subring.GenericSymbolicSubring

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

construction()¶

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)

has_valid_variable(variable)¶

Return whether the given variable is valid in this subring.

INPUT:

variable – a symbolic variable.

OUTPUT:

A 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

class sage.symbolic.subring.SymbolicSubringAcceptingVarsFunctor(vars)¶

Bases: sage.symbolic.subring.GenericSymbolicSubringFunctor

merge(other)¶

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

class sage.symbolic.subring.SymbolicSubringFactory¶

Bases: sage.structure.factory.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 (default: False) – a boolean. 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')

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))

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))

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))

create_key_and_extra_args(accepting_variables=None, rejecting_variables=None, no_variables=False, **kwds)¶

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

See SymbolicSubringFactory for details.

create_object(version, key, **kwds)¶

Create an object from the given arguments.

See SymbolicSubringFactory for details.

class sage.symbolic.subring.SymbolicSubringRejectingVars(vars)¶

Bases: sage.symbolic.subring.GenericSymbolicSubring

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

construction()¶

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)

has_valid_variable(variable)¶

Return whether the given variable is valid in this subring.

INPUT:

variable – a symbolic variable.

OUTPUT:

A 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

class sage.symbolic.subring.SymbolicSubringRejectingVarsFunctor(vars)¶

Bases: sage.symbolic.subring.GenericSymbolicSubringFunctor

merge(other)¶

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

Subrings of the Symbolic Ring¶

Classes and Methods¶

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