Tensor
# The Original Meanings of Words
* homo- : same
* iso- : equal
# Tensor
## Denotations
- $R$ : CRing
- $M,N$ : Mod(R)
## Definition : $( T , \otimes , M \times N )$ is Tensor
$(T,\otimes ,M\times N)\quad ;\quad\otimes : M\times N\rightarrow T\quad$ is Tensor.
========================================def
$\forall\ (X,\phi)\quad;\quad\phi : M\times N\rightarrow X$ : Bilinear map .
$\exists ! \ f : T \rightarrow X .$
$\hspace{4cm}$ $f\circ\otimes =\phi$
## Definition : Bilinear map $\phi$
$\phi : M \times N \rightarrow T$ is bilinear map
$\iff$
$\forall m \in M. \forall n \in M$ , $\phi$ is homomorphism on Additive Module
$\iff$
$$ \phi(m,n + n') = \phi(m,n) + \phi(m,n') $$
$$ \phi(m + m',n) = \phi(m,n) + \phi(m',n) $$
# Ring Homomorphism
Ring Homomorphism $f$
satisfies;
$$ f(n + m) = f(n) + f(m) $$
$$ f(n \bullet m) = f(n) \bullet f(m) \quad $$
$$ f(1) = 1 $$
# e.g. CRing(1-parametor Mod(R)) Tensor
CRing $R$ itself is one parametor $ Mod(R) $.
- $ R = \{0,1,2\} $ : CRing
- $ S = \{0,1,2\} $ : CRing
$$
\begin{array}{c|ccc}
+ & 0 & 1 & 2 \\
\hline
0 & 0 & 1 & 2 \\
1 & 1 & 2 & 0 \\
2 & 2 & 0 & 1
\end{array}
\quad
\begin{array}{c|ccc}
* & 0 & 1 & 2 \\
\hline
0 & 0 & 0 & 0 \\
1 & 0 & 1 & 2 \\
2 & 0 & 2 & 1
\end{array}
$$
1. $ \quad (0 \otimes 0) + (0 \otimes 0) = (0 \otimes 0) $
2. $ \quad (0 \otimes 1) + (1 \otimes 0) = (2 \otimes 1) + (1 \otimes 1) + (1 \otimes 0) = (0 \otimes 1) $
3. $ \quad (0 \otimes 1) + (1 \otimes 0) = (0 \otimes 1) + (1 \otimes 1) + (1 \otimes 2) = (1 \otimes 0) $
4. $ \quad (2 \otimes 0) + (2 \otimes 0) = (1 \otimes 0) $
5. $ \quad (2 \otimes 0) + (2 \otimes 0) = (2 \otimes 0) $
2,3,4,5 $ \rightarrow (0 \otimes 1) = (1 \otimes 0) = (0 \otimes 2) = (2 \otimes 0) $
$ (0 \otimes 0) = (1 \otimes 0) + (2 \otimes 0) = (2 \otimes 0) + (2 \otimes 0) = (1 \otimes 0) $
So we can define a Quotient Tensor like ;
$$ \bar{0} \quad := \quad (0 \otimes 0) \simeq (0 \otimes 1) \simeq (0 \otimes 2) \simeq (1 \otimes 0) \simeq (2 \otimes 0) $$
with simirarity ,
1. $ \quad (2 \otimes 1) + (2 \otimes 1) = (2 \otimes 2) $
2. $ \quad (2 \otimes 1) + (2 \otimes 1) = (1 \otimes 1) $
3. $ \quad (1 \otimes 1) + (1 \otimes 1) = (1 \otimes 2) $
4. $ \quad (1 \otimes 1) + (1 \otimes 1) = (2 \otimes 1) $
$$ \bar{1} \quad := \quad (1 \otimes 1) \simeq (2 \otimes 2) $$
$$ \bar{2} \quad := \quad (1 \otimes 2) \simeq (2 \otimes 1) $$
Thus,
Tensor also has Ring's Structure
$$
\begin{array}{c|ccc}
+ & \bar{0} & \bar{1} & \bar{2} \\
\hline
\bar{0} & \bar{0} & \bar{1} & \bar{2} \\
\bar{1} & \bar{1} & \bar{2} & \bar{0} \\
\bar{2} & \bar{2} & \bar{0} & \bar{1}
\end{array}
\quad
\begin{array}{c|ccc}
* & \bar{0} & \bar{1} & \bar{2} \\
\hline
\bar{0} & \bar{0} & \bar{0} & \bar{0} \\
\bar{1} & \bar{0} & \bar{1} & \bar{2} \\
\bar{2} & \bar{0} & \bar{2} & \bar{1}
\end{array}
$$