Studies of Euler diagrams/transformations

These are pairs of functions in the same clan (NP equivalence class), so they can be expressed in terms of each other. The clan numbers refer to the rational ordering. (Which will at some point be replaced by a better one.)

The transformation from one to the other is a, which means that arguments are negated and permuted. It can be just a set of negated places or just a permutation. (These cases are marked with N, P or NP respecitively.)

clan 84: and   (NP)
This Euler diagram has no symmetry. Therefore the transformation of one into the other is unique. The one from left to right is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{Blue}\neg 2} & {\color{Red}~0} & {\color{ForestGreen}\neg 1} & {\color{Orange}~3} \end{pmatrix} $$.  It means the following:
 * The old red border becomes the new blue border, and the orientation changes. (The spikes pointed upward, now they point downward.)
 * The old green border becomes the new red border. (The orientation stays the same.)
 * The old blue border becomes the new green border, and the orientation changes. (The spikes pointed left, now they point right.)
 * The yellow border remains unchanged.

The transformation from right to left is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{ForestGreen}~1} & {\color{Blue}\neg 2} & {\color{Red}\neg 0} & {\color{Orange}~3} \end{pmatrix} $$, the inverse of the one shown above.

clan 157: and   (NP)
is a gap variant of (shown above on the right).

These two functions do not have the same set of (relevant) arguments. But that is not a problem. One can see as a 5-ary function with an irrelevant input E, and  has an irrelevant input A.

The transformation from left to right is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} & {\color{Brown}~4} \\ {\color{Orange}\neg 3} & {\color{Blue}~2} & {\color{ForestGreen}~1} & {\color{Brown}~4} & {\color{Red}~0} \end{pmatrix} $$. It means the following:
 * The old red border becomes the new yellow border, and the orientation changes. (The spikes pointed left, now they point right.)
 * The old green border becomes the new blue border.
 * The old blue border becomes the new green border.
 * The old yellow border becomes the new brown border.
 * The inexistent old brown border "becomes" the inexistent new red border. (There could also be a negator in this place.)

The transformation from right to left is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} & {\color{Brown}~4} \\ {\color{Brown}~4} & {\color{Blue}~2} & {\color{ForestGreen}~1} & {\color{Red}\neg 0} & {\color{Orange}~3} \end{pmatrix} $$, the inverse of the one shown above.

clan 109: and   (NP)
The diagrams in this EC are mirror symmetric. That means that there are two transformations between each pair of functions. The one used here to get from left to right is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{Blue}\neg 2} & {\color{Orange}~3} & {\color{ForestGreen}~1} & {\color{Red}~0} \end{pmatrix} $$. The one from right to left is the inverse $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{Orange}~3} & {\color{Blue}~2} & {\color{Red}\neg 0} & {\color{ForestGreen}~1} \end{pmatrix} $$.

clan 203: and   (NP)
Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for. The transformation from the chosen diagram of to the one of  above is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{Red}~0} & {\color{Orange}~3} & {\color{Blue}\neg 2} & {\color{ForestGreen}\neg 1} \end{pmatrix} $$. Its inverse is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{Red}~0} & {\color{Orange}\neg 3} & {\color{Blue}\neg 2} & {\color{ForestGreen}~1} \end{pmatrix} $$.

The transformation from the chosen diagram of to the one of  below is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{ForestGreen}\neg 1} & {\color{Blue}\neg 2} & {\color{Orange}~3} & {\color{Red}~0} \end{pmatrix} $$. Its inverse is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} \\ {\color{Orange}~3} & {\color{Red}\neg 0} & {\color{ForestGreen}\neg 1} & {\color{Blue}~2} \end{pmatrix} $$.

This whole NP equivalence class with 192 functions (represented by 384 diagrams) can be found in NP.

clan 349: and   (P)
These diagrams have the symmetry of a rectangle, which can be flipped in four ways. This means that there are four transformations between each pair of functions in it. The one used here to get from left to right is $$ \begin{pmatrix} {\color{Red}0} & {\color{ForestGreen}1} & {\color{Blue}2} & {\color{Orange}3} \\ {\color{ForestGreen}1} & {\color{Blue}2} & {\color{Orange}3} & {\color{Red}0} \end{pmatrix} $$. The one from right to left is the inverse $$ \begin{pmatrix} {\color{Red}0} & {\color{ForestGreen}1} & {\color{Blue}2} & {\color{Orange}3} \\ {\color{Orange}3} & {\color{Red}0} & {\color{ForestGreen}1} & {\color{Blue}2} \end{pmatrix} $$.

The arguments are only permuted, but not negated. So only the colors change, but not the direction of the spikes.

clan 15: and   (NP)
Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for. The transformation between the chosen diagram of and the one of  above is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\ {\color{ForestGreen}\neg 1} & {\color{Red}\neg 0} & {\color{Blue}~2} \end{pmatrix} $$. (It is self-inverse, so it works left to right, and back.)

The transformation from the chosen diagram of to the one of  below is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\ {\color{Blue}\neg 2} & {\color{Red}~0} & {\color{ForestGreen}~1} \end{pmatrix} $$. Its inverse is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\ {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Red}\neg 0} \end{pmatrix} $$.

and  (N)
These two diagrams differ only in the orientation of border A. The self-inverse transformation between them is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\ {\color{Red}\neg 0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\ \end{pmatrix} $$.

and  (P)
The transformation between the two diagrams is $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} & {\color{Brown}~4} \\ {\color{Red}~0} & {\color{Orange}~3} & {\color{Brown}~4} & {\color{ForestGreen}~1} & {\color{Blue}~2} \end{pmatrix} $$, which is self-inverse. From left to right the columns 3 and 4 are irrelevant, and from right to left the columns 1 and 2. The entries in these columns could be negated and permuted. Counting these leads to 8 possible transformations. (That is between the two diagrams. Between the two functions it would be 16.)

clan 19: and   (N)
Euler diagrams in this EC have 3-fold. (This is obfuscated by the conventional representation on the right.) Thus there are six transformations from one function to another. The simplest one between these two is to change the orientation of border A, i.e. $$ \begin{pmatrix} {\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\ {\color{Red}\neg 0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\ \end{pmatrix} $$. (Compare potero and potula.)