Homogeneous ideals of free algebras.
For twosided ideals and when the base ring is a field, this implementation also provides Groebner bases and ideal containment tests.
EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: F
Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I
Twosided Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
One can compute Groebner bases out to a finite degree, can compute normal forms and can test containment in the ideal:
sage: I.groebner_basis(degbound=3)
Twosided Ideal (y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: (x*y*z*y*x).normal_form(I)
y*z*z*y*z + y*z*z*z*x + y*z*z*z*z
sage: x*y*z*y*x - (x*y*z*y*x).normal_form(I) in I
True
AUTHOR:
Bases: sage.rings.noncommutative_ideals.Ideal_nc
Graded homogeneous ideals in free algebras.
In the two-sided case over a field, one can compute Groebner bases up to a degree bound, normal forms of graded homogeneous elements of the free algebra, and ideal containment.
EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I
Twosided Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(2)
Twosided Ideal (x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(4)
Twosided Ideal (y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z, y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z, y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z, y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z, y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
Groebner bases are cached. If one has computed a Groebner basis out to a high degree then it will also be returned if a Groebner basis with a lower degree bound is requested:
sage: I.groebner_basis(2)
Twosided Ideal (y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z, y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z, y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z, y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z, y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
Of course, the normal form of any element has to satisfy the following:
sage: x*y*z*y*x - (x*y*z*y*x).normal_form(I) in I
True
Left and right ideals can be constructed, but only twosided ideals provide Groebner bases:
sage: JL = F*[x*y+y*z,x^2+x*y-y*x-y^2]; JL
Left Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: JR = [x*y+y*z,x^2+x*y-y*x-y^2]*F; JR
Right Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: JR.groebner_basis(2)
Traceback (most recent call last):
...
TypeError: This ideal is not two-sided. We can only compute two-sided Groebner bases
sage: JL.groebner_basis(2)
Traceback (most recent call last):
...
TypeError: This ideal is not two-sided. We can only compute two-sided Groebner bases
Also, it is currently not possible to compute a Groebner basis when the base ring is not a field:
sage: FZ.<a,b,c> = FreeAlgebra(ZZ, implementation='letterplace')
sage: J = FZ*[a^3-b^3]*FZ
sage: J.groebner_basis(2)
Traceback (most recent call last):
...
TypeError: Currently, we can only compute Groebner bases if the ring of coefficients is a field
The letterplace implementation of free algebras also provides integral degree weights for the generators, and we can compute Groebner bases for twosided graded homogeneous ideals:
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace',degrees=[1,2,3])
sage: I = F*[x*y+z-y*x,x*y*z-x^6+y^3]*F
sage: I.groebner_basis(Infinity)
Twosided Ideal (x*z*z - y*x*x*z - y*x*y*y + y*x*z*x + y*y*y*x + z*x*z + z*y*y - z*z*x,
x*y - y*x + z,
x*x*x*x*z*y*y + x*x*x*z*y*y*x - x*x*x*z*y*z - x*x*z*y*x*z + x*x*z*y*y*x*x +
x*x*z*y*y*y - x*x*z*y*z*x - x*z*y*x*x*z - x*z*y*x*z*x +
x*z*y*y*x*x*x + 2*x*z*y*y*y*x - 2*x*z*y*y*z - x*z*y*z*x*x -
x*z*y*z*y + y*x*z*x*x*x*x*x - 4*y*x*z*x*x*z - 4*y*x*z*x*z*x +
4*y*x*z*y*x*x*x + 3*y*x*z*y*y*x - 4*y*x*z*y*z + y*y*x*x*x*x*z +
y*y*x*x*x*z*x - 3*y*y*x*x*z*x*x - y*y*x*x*z*y +
5*y*y*x*z*x*x*x + 4*y*y*x*z*y*x - 4*y*y*y*x*x*z +
4*y*y*y*x*z*x + 3*y*y*y*y*z + 4*y*y*y*z*x*x + 6*y*y*y*z*y +
y*y*z*x*x*x*x + y*y*z*x*z + 7*y*y*z*y*x*x + 7*y*y*z*y*y -
7*y*y*z*z*x - y*z*x*x*x*z - y*z*x*x*z*x + 3*y*z*x*z*x*x +
y*z*x*z*y + y*z*y*x*x*x*x - 3*y*z*y*x*z + 7*y*z*y*y*x*x +
3*y*z*y*y*y - 3*y*z*y*z*x - 5*y*z*z*x*x*x - 4*y*z*z*y*x +
4*y*z*z*z - z*y*x*x*x*z - z*y*x*x*z*x - z*y*x*z*x*x -
z*y*x*z*y + z*y*y*x*x*x*x - 3*z*y*y*x*z + 3*z*y*y*y*x*x +
z*y*y*y*y - 3*z*y*y*z*x - z*y*z*x*x*x - 2*z*y*z*y*x +
2*z*y*z*z - z*z*x*x*x*x*x + 4*z*z*x*x*z + 4*z*z*x*z*x -
4*z*z*y*x*x*x - 3*z*z*y*y*x + 4*z*z*y*z + 4*z*z*z*x*x +
2*z*z*z*y,
x*x*x*x*x*z + x*x*x*x*z*x + x*x*x*z*x*x + x*x*z*x*x*x + x*z*x*x*x*x +
y*x*z*y - y*y*x*z + y*z*z + z*x*x*x*x*x - z*z*y,
x*x*x*x*x*x - y*x*z - y*y*y + z*z)
of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
Again, we can compute normal forms:
sage: (z*I.0-I.1).normal_form(I)
0
sage: (z*I.0-x*y*z).normal_form(I)
-y*x*z + z*z
Twosided Groebner basis with degree bound.
INPUT:
ASSUMPTIONS:
Currently, we can only compute Groebner bases for twosided
ideals, and the ring of coefficients must be a field. A
is raised if one of these conditions is violated.
NOTES:
EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
Since was cached and since its degree bound can not be
decreased, it may happen that, as a side effect of other tests,
it already has a degree bound bigger than 3. So, we can not
test against the output of I.groebner_basis():
sage: F.set_degbound(3)
sage: I.groebner_basis() # not tested
Twosided Ideal (y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(4)
Twosided Ideal (y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z, y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z, y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z, y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z, y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(2) is I.groebner_basis(4)
True
sage: G = I.groebner_basis(4)
sage: G.groebner_basis(3) is G
True
If a finite complete Groebner basis exists, we can compute it as follows:
sage: I = F*[x*y-y*x,x*z-z*x,y*z-z*y,x^2*y-z^3,x*y^2+z*x^2]*F
sage: I.groebner_basis(Infinity)
Twosided Ideal (z*z*z*y*y + z*z*z*z*x, z*x*x*x + z*z*z*y, y*z - z*y, y*y*x + z*x*x, y*x*x - z*z*z, x*z - z*x, x*y - y*x) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
Since the commutators of the generators are contained in the ideal, we can verify the above result by a computation in a polynomial ring in negative lexicographic order:
sage: P.<c,b,a> = PolynomialRing(QQ,order='neglex')
sage: J = P*[a^2*b-c^3,a*b^2+c*a^2]
sage: J.groebner_basis()
[b*a^2 - c^3, b^2*a + c*a^2, c*a^3 + c^3*b, c^3*b^2 + c^4*a]
Aparently, the results are compatible, by sending to
,
to
and
to
.
Reduction of this ideal by another ideal, or normal form of an algebra element with respect to this ideal.
INPUT:
OUTPUT:
EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I.reduce(F)
Twosided Ideal (0) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.reduce(x^3)
-y*z*x - y*z*y - y*z*z
sage: I.reduce([x*y])
Twosided Ideal (y*z, x*x - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.reduce(F*[x^2+x*y,y^2+y*z]*F)
Twosided Ideal (x*y + y*z, -y*x + y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
This function is an automatically generated C wrapper around the Singular function ‘NF’.
This wrapper takes care of converting Sage datatypes to Singular datatypes and vice versa. In addition to whatever parameters the underlying Singular function accepts when called this function also accepts the following keyword parameters:
INPUT:
args - a list of arguments
ring - a multivariate polynomial ring
execution of this function will interrupt the computation (default: True)
attributes assigned to Singular objects (default: None)
EXAMPLE:
sage: groebner = sage.libs.singular.ff.groebner
sage: P.<x, y> = PolynomialRing(QQ)
sage: I = P.ideal(x^2-y, y+x)
sage: groebner(I)
[x + y, y^2 - y]
sage: triangL = sage.libs.singular.ff.triang__lib.triangL
sage: P.<x1, x2> = PolynomialRing(QQ, order='lex')
sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
sage: I = Ideal(Ideal(f1,f2).groebner_basis()[::-1])
sage: triangL(I, attributes={I:{'isSB':1}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
[x2, x1^2],
[x2, x1^2],
[x2, x1^2]]
The Singular documentation for ‘NF’ is given below.
5.1.119 reduce
--------------
`*Syntax:*'
`reduce (' poly_expression`,' ideal_expression `)'
`reduce (' poly_expression`,' ideal_expression`,' int_expression
`)'
`reduce (' poly_expression`,' poly_expression`,' ideal_expression
`)'
`reduce (' vector_expression`,' ideal_expression `)'
`reduce (' vector_expression`,' ideal_expression`,' int_expression
`)'
`reduce (' vector_expression`,' module_expression `)'
`reduce (' vector_expression`,' module_expression`,'
int_expression `)'
`reduce (' vector_expression`,' poly_expression`,'
module_expression `)'
`reduce (' ideal_expression`,' ideal_expression `)'
`reduce (' ideal_expression`,' ideal_expression`,' int_expression
`)'
`reduce (' ideal_expression`,' matrix_expression`,'
ideal_expression `)'
`reduce (' module_expression`,' ideal_expression `)'
`reduce (' module_expression`,' ideal_expression`,' int_expression
`)'
`reduce (' module_expression`,' module_expression `)'
`reduce (' module_expression`,' module_expression`,'
int_expression `)'
`reduce (' module_expression`,' matrix_expression`,'
module_expression `)'
`reduce (' poly/vector/ideal/module`,' ideal/module`,' int`,'
intvec `)'
`reduce (' ideal`,' matrix`,' ideal`,' int `)'
`reduce (' poly`,' poly`,' ideal`,' int `)'
`reduce (' poly`,' poly`,' ideal`,' int`,' intvec `)'
`*Type:*'
the type of the first argument
`*Purpose:*'
reduces a polynomial, vector, ideal or module to its normal form
with respect to an ideal or module represented by a standard basis.
Returns 0 if and only if the polynomial (resp. vector, ideal,
module) is an element (resp. subideal, submodule) of the ideal
(resp. module). The result may have no meaning if the second
argument is not a standard basis.
The third (optional) argument of type int modifies the behavior:
* 0 default
* 1 consider only the leading term and do no tail reduction.
* 2 reduce also with bad ecart (in the local case)
* 4 reduce without division, return possibly a non-zero
constant multiple of the remainder
If a second argument `u' of type poly or matrix is given, the
first argument `p' is replaced by `p/u'. This works only for zero
dimensional ideals (resp. modules) in the third argument and
gives, even in a local ring, a reduced normal form which is the
projection to the quotient by the ideal (resp. module). One may
give a degree bound in the fourth argument with respect to a
weight vector in the fifth argument in order have a finite
computation. If some of the weights are zero, the procedure may
not terminate!
`*Note_*'
The commands `reduce' and `NF' are synonymous.
`*Example:*'
* Menu:
See
* ideal::
* module::
* std::
* vector::
This function is an automatically generated C wrapper around the Singular function ‘system’.
This wrapper takes care of converting Sage datatypes to Singular datatypes and vice versa. In addition to whatever parameters the underlying Singular function accepts when called this function also accepts the following keyword parameters:
INPUT:
args - a list of arguments
ring - a multivariate polynomial ring
execution of this function will interrupt the computation (default: True)
attributes assigned to Singular objects (default: None)
EXAMPLE:
sage: groebner = sage.libs.singular.ff.groebner
sage: P.<x, y> = PolynomialRing(QQ)
sage: I = P.ideal(x^2-y, y+x)
sage: groebner(I)
[x + y, y^2 - y]
sage: triangL = sage.libs.singular.ff.triang__lib.triangL
sage: P.<x1, x2> = PolynomialRing(QQ, order='lex')
sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
sage: I = Ideal(Ideal(f1,f2).groebner_basis()[::-1])
sage: triangL(I, attributes={I:{'isSB':1}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
[x2, x1^2],
[x2, x1^2],
[x2, x1^2]]
The Singular documentation for ‘system’ is given below.
5.1.141 system
--------------
`*Syntax:*'
`system (' string_expression `)'
`system (' string_expression`,' expression `)'
`*Type:*'
depends on the desired function, may be none
`*Purpose:*'
interface to internal data and the operating system. The
string_expression determines the command to execute. Some commands
require an additional argument (second form) where the type of the
argument depends on the command. See below for a list of all
possible commands.
`*Note_*'
Not all functions work on every platform.
`*Functions:*'
`system("sh"', string_expression `)'
shell escape, returns the return code of the shell as int.
The string is sent literally to the shell.
`system("pid")'
returns the process number as int (for creating unique names).
`system("--cpus")'
returns the number of cpu cores as int (for using multiple
cores).
`system("uname")'
returns a string identifying the architecture for which
SINGULAR was compiled.
`system("getenv",' string_expression`)'
returns the value of the shell environment variable given as
the second argument. The return type is string.
`system("setenv",'string_expression, string_expression`)'
sets the shell environment variable given as the second
argument to the value given as the third argument. Returns
the third argument. Might not be available on all platforms.
`system("tty")'
resets the terminal.
`system("version")'
returns the version number of SINGULAR as int.
`system("contributors")'
returns names of people who contributed to the SINGULAR
kernel as string.
`system("gen")'
returns the generating element of the multiplicative group of
(Z/p)\{0} (as int) where p is the characteristic of the
basering.
`system("nblocks")'
`system("nblocks",' ring_name `)'
returns the number of blocks of the given ring, or the number
of parameters of the current basering, if no second argument
is given. The return type is int.
`system("Singular")'
returns the absolute (path) name of the running SINGULAR as
string.
`system("SingularLib")'
returns the colon seperated library search path name as
string.
`system("'-`")'
prints the values of all options.
`system("'-long_option_name`")'
returns the value of the (command-line) option
long_option_name. The type of the returned value is either
string or int. *Note Command line options::, for more info.
`system("'-long_option_name`",' expression`)'
sets the value of the (command-line) option long_option_name
to the value given by the expression. Type of the expression
must be string, or int. *Note Command line options::, for
more info. Among others, this can be used for setting the
seed of the random number generator, the used help browser,
the minimal display time, or the timer resolution.
`system("browsers");'
returns a string about available help browsers. *Note The
online help system::. returns the number of cpus as int (for
creating multiple threads/processes).
`system("pid")'
`*Example:*'
// a listing of the current directory:
system("sh","ls");
// execute a shell, return to SINGULAR with exit:
system("sh","sh");
string unique_name="/tmp/xx"+string(system("pid"));
unique_name;
==> /tmp/xx4711
system("uname")
==> ix86-Linux
system("getenv","PATH");
==> /bin:/usr/bin:/usr/local/bin
system("Singular");
==> /usr/local/bin/Singular