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--- status: TODO
--- author(s):
--- notes:
-- TODO:
-- (tensor, Matrix, Matrix)
-- (tensor, Module, Module)
L := {
(tensor, QuotientRing, QuotientRing),
(tensor, PolynomialRing, PolynomialRing),
(tensor, QuotientRing, PolynomialRing),
(tensor, PolynomialRing, QuotientRing)
}
undocumented L
undocumented flatten table(L, {
DegreeLift, DegreeMap, DegreeRank, Degrees, Global, Heft,
Join, Local, MonomialOrder, MonomialSize, VariableBaseName,
Inverses, Variables, SkewCommutative, Weights, WeylAlgebra}, (m,o) -> [m, o])
doc ///
Node
Key
(tensor, Monoid, Monoid)
[(tensor, Monoid, Monoid), DegreeLift]
[(tensor, Monoid, Monoid), DegreeMap]
[(tensor, Monoid, Monoid), DegreeRank]
[(tensor, Monoid, Monoid), Degrees]
[(tensor, Monoid, Monoid), Global]
[(tensor, Monoid, Monoid), Heft]
[(tensor, Monoid, Monoid), Inverses]
[(tensor, Monoid, Monoid), Join]
[(tensor, Monoid, Monoid), Local]
[(tensor, Monoid, Monoid), MonomialOrder]
[(tensor, Monoid, Monoid), MonomialSize]
[(tensor, Monoid, Monoid), VariableBaseName]
[(tensor, Monoid, Monoid), Variables]
[(tensor, Monoid, Monoid), SkewCommutative]
[(tensor, Monoid, Monoid), Weights]
[(tensor, Monoid, Monoid), WeylAlgebra]
(tensor, Ring, Ring)
[(tensor, Ring, Ring), DegreeLift]
[(tensor, Ring, Ring), DegreeMap]
[(tensor, Ring, Ring), DegreeRank]
[(tensor, Ring, Ring), Degrees]
[(tensor, Ring, Ring), Global]
[(tensor, Ring, Ring), Heft]
[(tensor, Ring, Ring), Inverses]
[(tensor, Ring, Ring), Join]
[(tensor, Ring, Ring), Local]
[(tensor, Ring, Ring), MonomialOrder]
[(tensor, Ring, Ring), MonomialSize]
[(tensor, Ring, Ring), VariableBaseName]
[(tensor, Ring, Ring), Variables]
[(tensor, Ring, Ring), SkewCommutative]
[(tensor, Ring, Ring), Weights]
[(tensor, Ring, Ring), WeylAlgebra]
(symbol**, Ring, Ring)
(symbol**, Monoid, Monoid)
Headline
tensor product of monoids
Usage
C = A ** B
C = tensor(A, B, ...)
Inputs
A:{Ring,Monoid}
B:{Ring,Monoid}
DegreeRank => ZZ -- see @TO [monoid,DegreeRank]@
Degrees => List -- see @TO [monoid,Degrees]@
Inverses => Boolean -- see @TO [monoid,Inverses]@
Global => Boolean -- see @TO [monoid,Global]@
Local => Boolean -- see @TO [monoid,Local]@
MonomialOrder => List -- see @TO [monoid,MonomialOrder]@
MonomialSize => ZZ -- see @TO [monoid,MonomialSize]@
Variables => List -- see @TO [monoid,Variables]@
VariableBaseName => Symbol -- see @TO [monoid,VariableBaseName]@
Heft => List -- see @TO [monoid,Heft]@
Join => Boolean -- overrides the corresponding option in {\tt A}; see @TO [monoid,Join]@
DegreeMap => Boolean -- overrides the corresponding option in {\tt A}; see @TO [monoid,DegreeMap]@
DegreeLift => Boolean -- overrides the corresponding option in {\tt A}; see @TO [monoid,DegreeLift]@
SkewCommutative => Boolean -- ignored by this routine
Weights => List -- ignored by this routine
WeylAlgebra => List -- ignored by this routine
Outputs
C:{Ring,Monoid}
tensor product of the monoids or rings
Description
Text
This is the same as {\tt A ** B} except that options are allowed,
see @TO (symbol **, Monoid, Monoid)@ and @TO (symbol **, Ring, Ring)@.
This method allows many of the options available for monoids, see @TO monoid@ for details.
This method essentially combines the variables of {\tt A} and {\tt B} into one monoid or ring.
Example
kk = ZZ/101
A = kk[a,b]
B = kk[c,d,e]
Text
The simplest version is to simply use @TO symbol**@:
Example
describe(A**B)
Text
If you wish to change the variable names:
Example
describe tensor(A, B, VariableBaseName => p)
describe tensor(A, B, Variables => {a1,a2,b1,b2,b3})
Text
The tensor product of two singly graded rings is bigraded.
Sometimes you want a singly graded ring. Here is one way to get it:
Example
describe (C = tensor(A, B, DegreeRank => 1, Degrees => {5:1}))
degreeLength C
degreesRing C
Text
Packing monomials into smaller space is more efficient, but less flexible.
The default is 32 bits, so if you want to pack them into 8 bit exponents, use:
Example
describe tensor(A, B, MonomialSize => 8)
Text
The default monomial order for tensor products is a product order.
Sometimes other orders are more desirable, e.g. GRevLex, or an elimination order:
Example
describe (C = tensor(A, B, MonomialOrder => Eliminate numgens A))
describe (C = tensor(A, B, MonomialOrder => GRevLex))
Text
If you tensor two skew-commutative rings, (or one skew commutative ring with a commutative polynomial ring),
then all of the skew-commuting variables skew commute with each other:
Example
As = kk[a, b, SkewCommutative => true]
D = kk[c, d, e, SkewCommutative => true]
E = tensor(As, D)
describe E
c * a
Text
Similarly, tensoring two Weyl algebras (or one and a polynomial ring) produces
a Weyl algebra with both sets of non-commuting pairs.
Example
E = kk[x, Dx, WeylAlgebra => {x => Dx}]
tensor(E, E, Variables => {x, Dx, y, Dy})
describe oo
Text
Two polynomial rings must have the same coefficient ring, otherwise an error is issued.
Currently, there is no way to specify other rings over which to define the tensor product.
Example
A = ZZ/101[a, b]
B = A[x, y]
C = tensor(B, B, Variables => {x1, y1, x2, y2})
describe C
Text
The flat monoid with the all variables visible, including those from the base ring,
can be obtained as follows.
Example
C.FlatMonoid
Caveat
Not all of the options for monoid are useful here. Some are silently ignored.
SeeAlso
describe
degreesRing
degreeLength
symbol**
FlatMonoid
Node
Key
(tensor, RingMap, Matrix)
(tensor, RingMap, Module)
(symbol**, RingMap, Matrix)
(symbol**, RingMap, Module)
Headline
tensor product via a ring map
Usage
f ** M
tensor(f, M)
Inputs
f:RingMap
from $R \to S$
M:{Matrix,Module}
over the source ring $R$ of {\tt f}
Outputs
:{Matrix,Module}
the same type as {\tt M}, but over the target ring $S$ of {\tt f}
Description
Example
R = QQ[a..d]
S = QQ[s,t]
F = map(S,R,{s^4,s^3*t,s*t^3,t^4})
m = matrix{{a,b,c,d}}
F ** m
F ** image m
SeeAlso
(symbol SPACE, RingMap, Module)
(symbol SPACE, RingMap, Module)
Node
Key
(symbol**, List, List)
Headline
Cartesian product of two lists
Description
Example
{1, 2} ** {10, 20, 30}
///
|
