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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!

fricas
M:=FreeModule(Integer,Symbol)

\label{eq1}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{Symbol}\ })(1)
Type: Type
fricas
N:=FreeModule(Integer,Symbol)

\label{eq2}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{Symbol}\ })(2)
Type: Type
fricas
a1:='a1::M

\label{eq3}a 1(3)
Type: FreeModule(Integer,Symbol)
fricas
a2:='a2::M

\label{eq4}a 2(4)
Type: FreeModule(Integer,Symbol)
fricas
b1:='b1::N

\label{eq5}b 1(5)
Type: FreeModule(Integer,Symbol)
fricas
b2:='b2::N

\label{eq6}b 2(6)
Type: FreeModule(Integer,Symbol)
fricas
MxN:=TensorProduct(Integer,Symbol,Symbol,M,N);
Type: Type
fricas
t := tensor(a1 + a2, b1 + b2)$MxN;
Type: TensorProduct(Integer,Symbol,Symbol,FreeModule(Integer,Symbol),FreeModule(Integer,Symbol))
fricas
t

\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))
fricas
leadingMonomial t

\label{eq8}\left[ a 2, \: b 2 \right](8)
Type: Product(Symbol,Symbol)
fricas
numberOfMonomials t

\label{eq9}4(9)

Demonstrating the axioms of the tensor product:

fricas
x:M
Type: Void
fricas
y:M
Type: Void
fricas
u:M
Type: Void
fricas
p:=2*x+3*u

\label{eq10}{2 \  x}+{3 \  u}(10)
Type: FreeModule(Integer,Symbol)
fricas
q:=5*x+7*y+11*u

\label{eq11}{7 \  y}+{5 \  x}+{{11}\  u}(11)
Type: FreeModule(Integer,Symbol)
fricas
MxM:=TensorProduct(Integer,Symbol,Symbol,M,M);
Type: Type
fricas
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))
fricas
w:= 13*y+17*y+19*u

\label{eq13}{{30}\  y}+{{19}\  u}(13)
Type: FreeModule(Integer,Symbol)
fricas
test( tensor(p + q, w)$MxM = tensor(p, w)$MxM + tensor(q, w)$MxM )

\label{eq14} \mbox{\rm true} (14)
Type: Boolean
fricas
test( tensor(p, q + w)$MxM = tensor(p, q)$MxM + tensor(p, w)$MxM )

\label{eq15} \mbox{\rm true} (15)
Type: Boolean
fricas
test( tensor(p, 23*w)$MxM = 23*tensor(p, w)$MxM )

\label{eq16} \mbox{\rm true} (16)
Type: Boolean
fricas
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!




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