File:Inequivalent monotonic Boolean functions by actual arity, nominal arity 4.svg

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Image set Inequivalent monotonic 4-ary Boolean functions

The Sloane's 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 and if there is an and a such that .
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.

This SVG was created with Inkscape.
Image set Inequivalent monotonic Boolean functions by actual arity

Hasse diagram of the Sloane's A003182(n) inequivalent of monotonic Boolean functions of nominal arity n

The node color stands for the actual arity, e.g. red for 0 (constant) and yellow for 3.

The actual arity a (gray) corresponds to the ratio of monotonic functions to all functions in the BEC, which is 1/2a (blue).
This ratio is shown as an unreduced fraction over each node, so the sum of all the numerators is Dedekind(n).

The black index numbers in the nodes refer to the rational order of BECs.

This SVG was created with Inkscape.
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Watchduck
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