Fundamentals of AI

Fundamentals of AI
------------------

A Heiz Matrix (of order N) is an NxN matrix over the field F2, where N is a power of 2, and a minimum of N = 4.

These are often depicted as square grids, with 1 elements shaded, and 0 elements left blank.

The Heiz Monoid (of order N), is the set of all such matrices, with matrix multiplication as the monoid action.

There are a few matrices of special interest in AI. In the order 4 case:

/ 1 1 0 0 \    / 1 1 0 0 \    / 0 0 0 0 \
| 0 0 0 0 |    | 0 1 1 0 |    | 1 1 1 1 |
| 0 0 0 0 |    | 0 0 0 0 |    | 1 1 1 0 |
\ 0 0 0 0 /    \ 0 0 0 0 /    \ 0 1 0 0 /

     C              M              H

Observe that C, M^2 ∈ S ⊂ K

Where S is the sub-monoid  whose elements are equal to their own squares. ie, S = {A: A^2 = A}.

The