File:Inequivalent monotonic 4-ary Boolean functions.svg

The  A003182(4) = 30 clans of monotonic 4-ary Boolean functions ordered in a Hasse diagram

This is a selection from the Dedekind(4) = 168 monotonic functions, with each clan represented by one of its functions.

The function chosen as the representative of its clan is the one with the lowest ring count vector (RCV). The lowest RCV is unique for all clans except three with weights between 7 and 9: In clan 297 and its complement 312 all three functions have the same RCV. In the self-complementary clan 203 two of the twelve functions have the lowest RCV. (Compare this list.) The possible representatives of these clans are shown in the bottom right corner.

In this Hasse diagram there is an arrow between the clans ${\displaystyle A}$ and ${\displaystyle B}$ if there is an ${\displaystyle a\in A}$ and a ${\displaystyle b\in B}$ such that ${\displaystyle a\rightarrow b}$.
Generally this relation exists between the chosen representatives, exept for the three knots mentioned above.
For those with weight 7 and 9 the middle representative is chosen, so that the arrows to the lower (333→297) and from the upper (312→346) look intuitive, but not the arrows between them (297→203 and 203→312). For that with weight 8 the right representative is chosen, so the arrow from below looks intuitive (77→203) but not that to the right (203→92).

The index numbers refer to the rational order of clans.

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