Principalities and dominions of Boolean functions

Studies of Boolean functions

| truth tables  | Zhegalkin indices

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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.)

3-ary  | 4-ary noble

|

truth tables  | Ж
0001 0000  | 8  |  | 136  |  | 0000 0001 0001 0000  | 2176
0000 0100  | 32  |  | 160  |  | 0000 0001 0000 0100  | 8320
0000 0010  | 64  |  | 192  |  | 0000 0001 0000 0010  | 16512
0001 0100  | 40  |  | 40  |  | 0000 0000 0001 0100  | 10240
0001 0010  | 72  |  | 72  |  | 0000 0000 0001 0010  | 18432
0000 0110  | 96  |  | 96  |  | 0000 0000 0000 0110  | 24576

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.

Overview: 3-ary

* * *

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. Its quadrant is 2 (even and odious). Its faction has size 6 for arity 3 and size 24 for arity 4.

faction with quaestors

truth tables  | Ж  | quaestor
0100 1011  | 210  |  | 26  |  | 9
0010 1101  | 180  |  | 28  |
0110 0011  | 198  |  | 38  |  | 5
0011 1001  | 156  |  | 52  |
0110 0101  | 166  |  | 70  |  | 3
0101 1001  | 154  |  | 82  |

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).

As a single value these sets can be expressed as the quaestor dominion. (That is a pair of king index and quadrant.) For arity 3 it is (2, 2) and for arity 4 it is (8, 2).

Each faction can be assigned a quaestor dominion. Each great faction (a.k.a. squad) can be assigned a great quaestor dominion (which is only the king index).