Category:3-ary Boolean functions in octeract matrix; principalities and dominions

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2-ary 3-ary 4-ary principalities and dominions in hypercube matrix

Principalities are sets of 3-ary Boolean functions. Interpreted a indices of 4-ary nobles, they correspond to the factions of 4-ary nobles.

Dominions are closely related to them. The matrices on the right are the transposes of those on the left.
They are relevant, because each faction can be assigned a dominion.

Each row in the table below is denoted with a king index. (It is the index of a king among the 4-ary nobles.)
In the matrices on the left the king indices are highlighted in bold.

Each colored field in the matrices below contains not just an integer in 0...255, but also the corresponding 4-ary noble integer.
For the matrix on the left the two integers in each colored field are noble index and noble value. (E.g. 255 is the noble index of 65534.)

Truth tables that differ only in the least/most significant bit shall be called partners/friends.
Then the truth tables in great principalities contain complements and friends.
Their Zhegalkin indices contain partners and friends.
Great dominions contain complements and partners (both for truth tables and Zhegalkin indices). (They have that in common with half clusters.)

faction size	king king index	great principalities		great dominions
faction size	king king index	truth tables	Zhegalkin indices	truth tables	Zhegalkin indices
1	0 0
1	26752 104
3	1632 6
3	28384 110
4	5760 22
4	10920 42
6	2176 8
6	3808 14
12	680 2
	6856 26
	11464 44
overview