Monday 5 January 2015

Linguistic Note 11 Tensor Product / July 30 2007


Linguistic Note

11

Tensor Product


    TANAKA Akio

1            
Field     F
Linear space     VW
Additive group     X
Map    f : V × → X
F-bilinear map satisfies below condition.
α1, αV    βW
 α1 + α2β) = f (α1, β ) + f (α2, β )
αV    β1 , β2W
αβ1 + β2) = f (αβ1) + (αβ 2)
αV    βW  λ∈ F
(λαβ) = f (α ,λβ )

2
Field     F
Linear space     VW
Additive group     T
F-bilinear map    τ BL ( V ×WT )
F-bilinear map     f  BL (V×W, X )
Additive group’s homomorphism      f T  X
Tensor product of V and W satisfies below condition.
Pair ( Tτ)
f = f ~ . τ

3
Tensor product V      ( Tτ)
Tensor product W      (  `τ` )
Additive isomorphism      φ : T  T `
τ` = τ . φ

4
Tensor product is expressed by the following briefly.
V  W

5
Algebra on field F     AB
Tensor product  B is defined by the following.
xy   B
xy : = ( mA   mB ) ( h ( x   ) )
h : = ( A  B )  ( A  B )  ( A  A ) ( B  B )
mA   m: ( A   A )  (   B )  A   B

6
Algebra on field F      A, B, C
 Hom Fal ( AC )
 Hom Fal ( BC )
α   A     β B
(α) g(β) = g (β(α)
h (α  β ) = (αg (β)

[Note]
Relationship between bilinear map τ , τ ` and isomorphism φ , namely τ` = τ . φ, may be helpful to word ( τ ) , sentence ( τ` ) and grammar ( φ ).

[Reference]
Frame-Quantum Theory     Tokyo March 13, 2005
Frame-Quantum Theory Addendum     Tokyo March 26, 2005
Frame-Quantum Theory map 3     Tokyo Hakuba March 28, 2005
Compendium     Premise for Frame-Quantum Theory     Tokyo March 22 – April 10, 2005

Tokyo July 30 2007
Sekinan Research Field of Language

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