MathAction SandBoxTensorProduct

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Date: Fri, 15 May 2009 21:03:00 +0200 Franz Lehner wrote:

Attached is a prototype for tensor products. It is free modules over commutative rings.

TensorProduct? is now included with FriCAS. Thank you Franz!

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M:=FreeModule(Integer,Symbol)

$\label{eq1}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{Symbol}\ })$

(1)

Type: Type

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N:=FreeModule(Integer,Symbol)

$\label{eq2}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{Symbol}\ })$

(2)

Type: Type

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a1:='a1::M

$\label{eq3}a 1$

(3)

Type: FreeModule(Integer,Symbol)

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a2:='a2::M

$\label{eq4}a 2$

(4)

Type: FreeModule(Integer,Symbol)

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b1:='b1::N

$\label{eq5}b 1$

(5)

Type: FreeModule(Integer,Symbol)

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b2:='b2::N

$\label{eq6}b 2$

(6)

Type: FreeModule(Integer,Symbol)

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MxN:=TensorProduct(Integer,Symbol,Symbol,M,N);

Type: Type

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t := tensor(a1 + a2, b1 + b2)$MxN;

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule(Integer,Symbol),FreeModule(Integer,Symbol))

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$\label{eq7}{a 2 \otimes b 2}+{a 2 \otimes b 1}+{a 1 \otimes b 2}+{a 1 \otimes b 1}$

(7)

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule(Integer,Symbol),FreeModule(Integer,Symbol))

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leadingMonomial t

$\label{eq8}a 2 \otimes b 2$

(8)

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule(Integer,Symbol),FreeModule(Integer,Symbol))

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numberOfMonomials t

$\label{eq9}4$

(9)

Type: PositiveInteger?

Demonstrating the axioms of the tensor product:

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x:M

Type: Void

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y:M

Type: Void

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u:M

Type: Void

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p:=2*x+3*u

$\label{eq10}{2 \ x}+{3 \ u}$

(10)

Type: FreeModule(Integer,Symbol)

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q:=5*x+7*y+11*u

$\label{eq11}{7 \ y}+{5 \ x}+{{11}\ u}$

(11)

Type: FreeModule(Integer,Symbol)

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MxM:=TensorProduct(Integer,Symbol,Symbol,M,M);

Type: Type

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r := tensor(p, q)$MxM

$\label{eq12}\begin{array}{@{}l} \displaystyle {{14}\ {x \otimes y}}+{{10}\ {x \otimes x}}+{{22}\ {x \otimes u}}+{{21}\ {u \otimes y}}+ \ \ \displaystyle {{15}\ {u \otimes x}}+{{33}\ {u \otimes u}}$

(12)

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule(Integer,Symbol),FreeModule(Integer,Symbol))

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w:= 13*y+17*y+19*u

$\label{eq13}{{30}\ y}+{{19}\ u}$

(13)

Type: FreeModule(Integer,Symbol)

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test( tensor(p + q, w)$MxM = tensor(p, w)$MxM + tensor(q, w)$MxM )

$\label{eq14} \mbox{\rm true}$

(14)

Type: Boolean

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test( tensor(p, q + w)$MxM = tensor(p, q)$MxM + tensor(p, w)$MxM )

$\label{eq15} \mbox{\rm true}$

(15)

Type: Boolean

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test( tensor(p, 23*w)$MxM = 23*tensor(p, w)$MxM )

$\label{eq16} \mbox{\rm true}$

(16)

Type: Boolean

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test( tensor(23*p, w)$MxM = 23*tensor(p, w)$MxM )

$\label{eq17} \mbox{\rm true}$

(17)

Type: Boolean

first example of a bialgebra --Bill Page, Wed, 13 May 2009 08:06:57 -0700 reply

SandBoxHopfAlgebra

TensorProduct? is now included with FriCAS. Thank you Franz!

Subject: Be Bold !!

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