File:Z2^3; Lattice of subgroups Hasse diagram adjacency matrix.svg
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Summary
Among the 8x8 matrices in files like 
there are sixteen, that describe equivalence relations.
(The power of two is a coincidence. There are five 4x4 matrices of this kind, and sixty seven 16x16 matrices.)
This matrix shows, how they are ordered by the refinement of relation,
and thus it's substantially equivalent to a Hasse diagram:
The strong red fields correspond to the edges in a Hasse diagram.
There are 35 strong red fields and 66 red fields in all.
This matrix is symmetrical to the subdiagonal.
The small black numbers over the 8x8 matrices are their bottom rows, read as binary numbers. Their numerical order justifies the bigger gray numbers, starting from 0, which simply denote the matrices' position in this sequence.
These 8x8 matrices' top rows:
1 0 0 0 0 0 0 0 (1)
1 1 0 0 0 0 0 0 (2)
1 0 1 0 0 0 0 0 (2)
1 0 0 1 0 0 0 0 (2)
1 1 1 1 0 0 0 0 (4)
1 0 0 0 1 0 0 0 (2)
1 0 0 0 0 1 0 0 (2)
1 1 0 0 1 1 0 0 (4)
1 0 0 0 0 0 1 0 (2)
1 0 1 0 1 0 1 0 (4)
1 0 0 1 0 1 1 0 (4)
1 0 0 0 0 0 0 1 (2)
1 0 0 1 1 0 0 1 (4)
1 0 1 0 0 1 0 1 (4)
1 1 0 0 0 0 1 1 (4)
1 1 1 1 1 1 1 1 (8)
(The binary digit sum is shown in parentheses.)
The elements can be ordered in different ways. This order is probably the best one.
The 8x8 matrix in the top right corner shows XNOR operations of arguments in lexicographical order.
(Compare the description of the bigger equivalent of this matrix.)
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 07:14, 26 July 2014 | | 1,600 × 1,585 (1.13 MB) | Wereldburger758 | Saved as plain SVG. Valid SVG now. |
| 20:48, 12 March 2010 |  | 1,600 × 1,585 (1.38 MB) | Mate2Code | {{Information |Description=This matrix shows the same relation as File:Boolean partition lattice 3.svg and File:Boolean partition lattice 3 rearranged by digit sum.svg, but the elements are rearranged in a different order, so that the 8x8 [[w:en |
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