notes.txt

N=2 -> 2
    1 0    0 1
    0 0    1 1

    just relabellings of each other

N=3 -> 0
N=4 -> 24

  direct product of (N=2) x (N=2) looks like
    3 2  1 0
    2 2  0 0
   
    1 0  1 0
    0 0  0 0

  this has swap(1,2) symmetry, so there are 4!/2! = 12 equivalent labellings

  this entry is a central groupoid (satisfies eq168)
    0 0 1 1
    2 2 3 3
    0 0 1 1
    2 2 3 3
  which has has symmetry to relabelling: 
    (0, 1, 2, 3) -> (3, 2, 1, 0)
  so again there are 12 copies of this

N=5 -> 0
N=6 -> 0
N=7 -> 0
N=8 -> ??  stopped after [0,0] -> 2
  no results with [0,0] = 0
  35520 with [0,0] = 1

  that fixes one, so 7! = 5040 copies of an entry with no symmetry

  expect 7!/2 = 2520 of (N=2)x(N=4)
  expect 7!/6 = 840 of (N=2)x(N=2)x(N=2)

  ?? leaves a lot

  can at least say:
    - there is no 8x8 with an idempotent element
    - picking some results that looked possibly unique, here is one with no symmetry
        [[1, 0, 0, 1, 2, 2, 3, 3],
         [0, 0, 0, 0, 3, 3, 3, 3],
         [4, 4, 5, 4, 5, 5, 4, 5],
         [7, 4, 5, 7, 6, 6, 4, 5],
         [1, 0, 0, 1, 6, 6, 3, 3],
         [7, 4, 5, 7, 2, 2, 4, 5],
         [4, 4, 5, 4, 5, 5, 4, 5],
         [0, 0, 0, 0, 3, 3, 3, 3]]

      it is distinct from (N=2)x(N=2)x(N=2)
      which will have one element that left and right multiplies to a constant

      and now 6 fold symmetry (permuting between N=2 copies)
      perm: (0, 1, 2, 3, 4, 5, 6, 7)
      perm: (0, 1, 2, 3, 6, 7, 4, 5)
      perm: (0, 1, 4, 5, 2, 3, 6, 7)
      perm: (0, 1, 4, 5, 6, 7, 2, 3)
      perm: (0, 1, 6, 7, 2, 3, 4, 5)
      perm: (0, 1, 6, 7, 4, 5, 2, 3)
        ((1, 0, 3, 2, 5, 4, 7, 6),
         (0, 0, 0, 0, 0, 0, 0, 0),    <--- presuming this is (N=2)x(N=2)x(N=2)
         (3, 0, 3, 0, 0, 3, 0, 3),
         (2, 0, 0, 2, 5, 7, 7, 5),
         (5, 0, 0, 5, 5, 0, 0, 5),
         (4, 0, 3, 7, 0, 4, 7, 3),
         (7, 0, 0, 7, 0, 7, 7, 0),
         (6, 0, 3, 5, 5, 3, 0, 6))

      it is distinct from (N=2)x(N=4 central groupoid)
      as that has some symmetry
      (0, 1, 2, 3, 4, 5, 6, 7) -> (3, 2, 1, 0, 7, 6, 5, 4)

         4 4 5 5  0 0 1 1
         6 6 7 7  2 2 3 3
         4 4 5 5  0 0 1 1
         6 6 7 7  2 2 3 3
        
         0 0 1 1  0 0 1 1
         2 2 3 3  2 2 3 3
         0 0 1 1  0 0 1 1
         2 2 3 3  2 2 3 3