
$$\begin{array}{r c l}{{z_{o_{N}}\,w=w_{o_{N}}\,z}}&{{=}}&{{\sum_{r=1}^{k-1}}}&{{z_{i_{1}}\cdots z_{i_{r}}\,w_{j_{r+1}}\cdots w_{j_{k}}}}\\ {{}}&{{}}&{{\{i_{1},\ldots,i_{r},j_{r+1},\ldots,j_{k}\}\equiv\{1,2,\ldots,k\}}}\\ {{}}&{{}}&{{\binom{i_{1}\ldots i_{r}}{i_{1}\ldots i_{r}}\in C^{r}(1,2,\ldots,k)}}\\ {{}}&{{}}&{{\binom{j_{r+1}\ldots,j_{k}}{\in C^{k-r}(1,2,\ldots,k)}}}\end{array}$$

$${\mathrm{~the~elements~}}\left\{1,2,...,k\right\}{\mathrm{~to~}}$$

where (1,2,..., ) r C k means the set of combinations of the elements { } 1, 2,..., k taken by r . 

[Similarly for (1,2,..., ) k r C k − ]. 

 In other words, No z w is the sum of all possible products of the components of vectors z and w , such that each product has at least a i z factor and at least wj factor, and each product has exactly k factors where each factor is a different vector component of z or of w . Similarly if we multiply three vectors: 

$$T_{o_{N}}I_{o_{N}}F=$$
C kl l C k − − + + − k 2 ∑ TIF T I F F + + + + ... N N u j j uv k u uv u u uv uv k u u u uv o o ii l l uvk u v i ij j l l k i i C kj j ...  , , 1  ,..., , ,..., , ,..., 1,2,...,  ,..., 1,2,..., , ,..., 1 ... 1 1 + + −−= ≡ ∈ ∈ { } { } ( )( ) ( ) 11 1 1 1 + + ++  1,2,..., , ,..., 1,2,..., v kuv uv k ∈ ∈ ( ) ( ) 1 ( )

 

Let's see an example for k = 3. 
$$\begin{array}{c}{{-\,{\cal S}\,.}}\\ {{\quad x_{1}\left(T_{1},I_{1},F_{1}\right)}}\\ {{\quad x_{2}\left(T_{2},I_{2},F_{2}\right)}}\end{array}$$
2 22 2 x TIF ( ) , , 3 33 3 123 123 1 2 3 , , F NN N T T TTT I I I I I F FF F oo o == = 123 123 123 123 123 123 N T I TI I IT I I I T TT I TI T ITT o =+++++ 1 2 3 12 3 1 23 12 3 1 23 123 N T F TF F FT F FFT TT F TFT FTT o = +++++ 1 2 3 12 3 1 23 12 3 1 23 123 No I F I FF FI F FFI I I F I FI FI I = +++++ 12 3 1 23 12 3 1 23 123 123 N N T I F TI F TF I I T F I FT FI T FT I o o =+++++
 For the case when indeterminacy I is not decomposed in subcomponents {as for example I = ∪P U where P =paradox (true and false simultaneously) and U =uncertainty (true or false, not sure which one)}, the previous formulas can be easily written using only three components as: 

$$T_{o_{_N}}I_{o_{_N}}F=\sum_{i,j,r\in P(1,2,3)}T_{i}I_{j}F_{r}$$
where P(1,2,3) means the set of permutations of (1,2,3) i.e. 
{ } (1, 2,3),(1,3, 2),(2,1,3),(2,3,1,),(3,1, 2),(3, 2,1)
2 3 = + ∑ z w zww wz z = ≡ ∈  i1 ( , , ) (1,2,3) ( , ) (1,2,3) N r o i j j i jr i jr j r P This neurotrophic law is associative and commutative. 

## Neutrophic Logic Operators
