Principalities and dominions of Boolean functions
Studies of Boolean functions |
truth tables | Zhegalkin indices | |
---|---|---|
principality | ||
dominion |
A principality is a set of n-ary truth tables whose (n+1)-ary noble equivalents form a faction.
Dominions are closely related to them. The reversed truth tables of the principalities are the Zhegalkin indices of the dominion.
The following table shows the six members of the red principality, which are also shown in the matrices on the right.
A representative of the 4-ary noble faction can be seen in the bottom left corner of this image.
(It is easily seen, that this tetrahedron can be permuted into six different positions.)
Overviews on Commons:
2-ary,
3-ary,
4-ary
Interesting subsets are those with entries on the diagonal and in the top row of the matrix of Zhegalkin indices.
Principalities follow directly from the concept of nobles. Dominions are less obvious, but arguably more important.
The quaestors of each faction form a dominion. Many factions with the same quadrant belong to the same dominion.
Take 0100 1011 (Ж 26) as an example. (It has quadrant 2).
For arity 3 the quaestors are {3, 5, 9} (yellow in this file)
and for arity 4 they are {15, 51, 85, 153, 165, 195} (yellow in this file, also shown above in the bottom left corner).