**MATRIX LOGIC**

All boolean functions are odd-linear, either way that you map
logic to Z mod 2! This opens up all the mechanics of linear algebra, including vectors
and matrices. This 'matrix logic' has two peculiarities; all sums must have an
odd number of terms, and there are several multiplications, two of them
versions of addition!

Define these matrix logic operations:

x

a b c
and y
= (a and x) +(b and y)+(c and
z)

z

x

a b
c or y
= (a or x) +(b or y)+(c or z)

z

x

a b
c --> y
= (a --> x) +(b --> y)+(c
--> z)

z

x

a b
c - y
= (a - x) +(b - y)+(c - z)

z

x

a b
c iff y =
(a iff x) +(b iff y)+(c iff z)

z

x

a b
c xor y
= (a xor x) +(b xor y)+(c xor
z)

z

That is, we do "dot products" with arbitrary logic
operators as

multiplications. Extend this
to odd-by-odd products.

For instance, the 'matrix and'

a11 a12 a13
b11 b12 b13

a21 a22 a23
and b21
b22 b23

a31 a32 a33
b31 b32
b33

is a 3 by 3 logic matrix with i-j entry equal to:

(ai1 and b1j) + (ai2 and b2j) + (ai3 and b3j)

Define 'matrix or' similarly; along with matrix ->, -,
etc.

There are matrix scalars too:

a and (
x y
z ) =
( (a and x) (a and y)
(a and z) )

a xor (
x y
z ) = ( (a xor x)
(a xor y) (a xor
z) )

and so on.

Matrix 'and' and 'or' are associative; but of course
commutativity fails, and I am not sure about distribution of 'and' and 'or'
over each other.

Matrix 'and' has this identity matrix:

T F F

F T F

F F T

Matrix 'or' has this identity matrix:

F T T

T F T

T T F

The "and-determinant", And-det, of

a b c

d e f

g h i

is

(a and e and i) xor (b and f
and g) xor (c and d and h) xor

(g and e and c) xor (h and f
and a) xor (i and d and b)

Equivalently; rewrite the matrix with F=0 and T=1, find the
determinant mod 2, then translate back.

Given logic matrices A and B:

And-det( A and B )
= And-det(A) and And-det(B)

The matrix has an and-inverse if its and-determinant equals
T.

The "or-determinant" Or-det of

a b c

d e f

g h i

is

(a or e or i) iff (b or f or
g) iff (c or d or h) iff (g or e or c) iff

(h or f or a) iff (i or d or
b)

Equivalently; rewrite the matrix with F=1 and T=0, find the
determinant mod 2, then translate back.

Given logic matrices A and B:

Or-det( A or B )
= Or-det(A) or Or-det(B)

The matrix has an or-inverse if its or-determinant equals F.

You can of course define eigenvectors and eigenvalues. In
fact you get the odd-by-odd matrices over Z mod 2, with both 'and' and 'or' multiplications,
along with -->, -, xor and iff!

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