File:Z2^3; Lattice of subgroups Hasse diagram adjacency matrix.svg

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.)

The SVG code is valid.   This vector image was created with Inkscape by v .

This SVG file was uploaded with Commonist.

I, the copyright holder of this work, hereby publish it under the following licenses:

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

This file is licensed under the Creative Commons Attribution 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.

https://creativecommons.org/licenses/by/3.0CC BY 3.0 Creative Commons Attribution 3.0 truetrue

You may select the license of your choice.