Weighted homogeneous elements of free algebras, in letterplace implementation

AUTHOR:

class sage.algebras.letterplace.free_algebra_element_letterplace.FreeAlgebraElement_letterplace[source]

Bases: AlgebraElement

Weighted homogeneous elements of a free associative unital algebra (letterplace implementation).

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: x+y
x + y
sage: x*y !=y*x
True
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: (y^3).reduce(I)
y*y*y
sage: (y^3).normal_form(I)
y*y*z - y*z*y + y*z*z

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> x+y
x + y
>>> x*y !=y*x
True
>>> I = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F
>>> (y**Integer(3)).reduce(I)
y*y*y
>>> (y**Integer(3)).normal_form(I)
y*y*z - y*z*y + y*z*z

Here is an example with nontrivial degree weights:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: I = F*[x*y-y*x, x^2+2*y*z, (x*y)^2-z^2]*F
sage: x.degree()
2
sage: y.degree()
1
sage: z.degree()
3
sage: (x*y)^3
x*y*x*y*x*y
sage: ((x*y)^3).normal_form(I)
z*z*y*x
sage: ((x*y)^3).degree()
9

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(2),Integer(1),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y-y*x, x**Integer(2)+Integer(2)*y*z, (x*y)**Integer(2)-z**Integer(2)]*F
>>> x.degree()
2
>>> y.degree()
1
>>> z.degree()
3
>>> (x*y)**Integer(3)
x*y*x*y*x*y
>>> ((x*y)**Integer(3)).normal_form(I)
z*z*y*x
>>> ((x*y)**Integer(3)).degree()
9
degree()[source]

Return the degree of this element.

Note

Generators may have a positive integral degree weight. All elements must be weighted homogeneous.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((x+y+z)^3).degree()
3
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((x*y+z)^3).degree()
9

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((x+y+z)**Integer(3)).degree()
3
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(2),Integer(1),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((x*y+z)**Integer(3)).degree()
9
lc()[source]

The leading coefficient of this free algebra element, as element of the base ring.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lc()
20
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lc().parent() is F.base()
True
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2).lc()
4

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((Integer(2)*x+Integer(3)*y-Integer(4)*z)**Integer(2)*(Integer(5)*y+Integer(6)*z)).lc()
20
>>> ((Integer(2)*x+Integer(3)*y-Integer(4)*z)**Integer(2)*(Integer(5)*y+Integer(6)*z)).lc().parent() is F.base()
True
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(2),Integer(1),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((Integer(2)*x*y+z)**Integer(2)).lc()
4
letterplace_polynomial()[source]

Return the commutative polynomial that is used internally to represent this free algebra element.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((x+y-z)^2).letterplace_polynomial()
x*x_1 + x*y_1 - x*z_1 + y*x_1 + y*y_1 - y*z_1 - z*x_1 - z*y_1 + z*z_1

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((x+y-z)**Integer(2)).letterplace_polynomial()
x*x_1 + x*y_1 - x*z_1 + y*x_1 + y*y_1 - y*z_1 - z*x_1 - z*y_1 + z*z_1

If degree weights are used, the letterplace polynomial is homogenized by slack variables:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((x*y+z)^2).letterplace_polynomial()
x*x__1*y_2*x_3*x__4*y_5 + x*x__1*y_2*z_3*x__4*x__5 + z*x__1*x__2*x_3*x__4*y_5 + z*x__1*x__2*z_3*x__4*x__5

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(2),Integer(1),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((x*y+z)**Integer(2)).letterplace_polynomial()
x*x__1*y_2*x_3*x__4*y_5 + x*x__1*y_2*z_3*x__4*x__5 + z*x__1*x__2*x_3*x__4*y_5 + z*x__1*x__2*z_3*x__4*x__5
lm()[source]

The leading monomial of this free algebra element.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lm()
x*x*y
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2).lm()
x*y*x*y

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((Integer(2)*x+Integer(3)*y-Integer(4)*z)**Integer(2)*(Integer(5)*y+Integer(6)*z)).lm()
x*x*y
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(2),Integer(1),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((Integer(2)*x*y+z)**Integer(2)).lm()
x*y*x*y
lm_divides(p)[source]

Tell whether or not the leading monomial of self divides the leading monomial of another element.

Note

A free algebra element \(p\) divides another one \(q\) if there are free algebra elements \(s\) and \(t\) such that \(spt = q\).

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2*z).lm()
x*y*x*y*z
sage: (y*x*y-y^4).lm()
y*x*y
sage: (y*x*y-y^4).lm_divides((2*x*y+z)^2*z)
True

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(2),Integer(1),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((Integer(2)*x*y+z)**Integer(2)*z).lm()
x*y*x*y*z
>>> (y*x*y-y**Integer(4)).lm()
y*x*y
>>> (y*x*y-y**Integer(4)).lm_divides((Integer(2)*x*y+z)**Integer(2)*z)
True
lt()[source]

The leading term (monomial times coefficient) of this free algebra element.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lt()
20*x*x*y
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2).lt()
4*x*y*x*y

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((Integer(2)*x+Integer(3)*y-Integer(4)*z)**Integer(2)*(Integer(5)*y+Integer(6)*z)).lt()
20*x*x*y
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(2),Integer(1),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> ((Integer(2)*x*y+z)**Integer(2)).lt()
4*x*y*x*y
normal_form(I)[source]

Return the normal form of this element with respect to a twosided weighted homogeneous ideal.

INPUT:

  • I – a twosided homogeneous ideal of the parent \(F\) of this element, \(x\)

OUTPUT: the normal form of \(x\) wrt. \(I\)

Note

The normal form is computed by reduction with respect to a Groebnerbasis of \(I\) with degree bound \(deg(x)\).

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: (x^5).normal_form(I)
-y*z*z*z*x - y*z*z*z*y - y*z*z*z*z

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F
>>> (x**Integer(5)).normal_form(I)
-y*z*z*z*x - y*z*z*z*y - y*z*z*z*z

We verify two basic properties of normal forms: The difference of an element and its normal form is contained in the ideal, and if two elements of the free algebra differ by an element of the ideal then they have the same normal form:

sage: x^5 - (x^5).normal_form(I) in I
True
sage: (x^5+x*I.0*y*z-3*z^2*I.1*y).normal_form(I) == (x^5).normal_form(I)
True

>>> from sage.all import *
>>> x**Integer(5) - (x**Integer(5)).normal_form(I) in I
True
>>> (x**Integer(5)+x*I.gen(0)*y*z-Integer(3)*z**Integer(2)*I.gen(1)*y).normal_form(I) == (x**Integer(5)).normal_form(I)
True

Here is an example with non-trivial degree weights:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[1,2,3])
sage: I = F*[x*y-y*x+z, y^2+2*x*z, (x*y)^2-z^2]*F
sage: ((x*y)^3).normal_form(I)
z*z*y*x - z*z*z
sage: (x*y)^3-((x*y)^3).normal_form(I) in I
True
sage: ((x*y)^3+2*z*I.0*z+y*I.1*z-x*I.2*y).normal_form(I) == ((x*y)^3).normal_form(I)
True

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', degrees=[Integer(1),Integer(2),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y-y*x+z, y**Integer(2)+Integer(2)*x*z, (x*y)**Integer(2)-z**Integer(2)]*F
>>> ((x*y)**Integer(3)).normal_form(I)
z*z*y*x - z*z*z
>>> (x*y)**Integer(3)-((x*y)**Integer(3)).normal_form(I) in I
True
>>> ((x*y)**Integer(3)+Integer(2)*z*I.gen(0)*z+y*I.gen(1)*z-x*I.gen(2)*y).normal_form(I) == ((x*y)**Integer(3)).normal_form(I)
True
reduce(G)[source]

Reduce this element by a list of elements or by a twosided weighted homogeneous ideal.

INPUT:

Either a list or tuple of weighted homogeneous elements of the free algebra, or an ideal of the free algebra, or an ideal in the commutative polynomial ring that is currently used to implement the multiplication in the free algebra.

OUTPUT: the twosided reduction of this element by the argument

Note

This may not be the normal form of this element, unless the argument is a twosided Groebner basis up to the degree of this element.

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: p = y^2*z*y^2+y*z*y*z*y

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F
>>> p = y**Integer(2)*z*y**Integer(2)+y*z*y*z*y

We compute the letterplace version of the Groebner basis of \(I\) with degree bound 4:

sage: G = F._reductor_(I.groebner_basis(4).gens(),4)
sage: G.ring() is F.current_ring()
True

>>> from sage.all import *
>>> G = F._reductor_(I.groebner_basis(Integer(4)).gens(),Integer(4))
>>> G.ring() is F.current_ring()
True

Since the element \(p\) is of degree 5, it is no surprise that its reductions with respect to the original generators of \(I\) (of degree 2), or with respect to \(G\) (Groebner basis with degree bound 4), or with respect to the Groebner basis with degree bound 5 (which yields its normal form) are pairwise different:

sage: p.reduce(I)
y*y*z*y*y + y*z*y*z*y
sage: p.reduce(G)
y*y*z*z*y + y*z*y*z*y - y*z*z*y*y + y*z*z*z*y
sage: p.normal_form(I)
y*y*z*z*z + y*z*y*z*z - y*z*z*y*z + y*z*z*z*z
sage: p.reduce(I) != p.reduce(G) != p.normal_form(I) != p.reduce(I)
True

>>> from sage.all import *
>>> p.reduce(I)
y*y*z*y*y + y*z*y*z*y
>>> p.reduce(G)
y*y*z*z*y + y*z*y*z*y - y*z*z*y*y + y*z*z*z*y
>>> p.normal_form(I)
y*y*z*z*z + y*z*y*z*z - y*z*z*y*z + y*z*z*z*z
>>> p.reduce(I) != p.reduce(G) != p.normal_form(I) != p.reduce(I)
True