Seal (discrete mathematics)

Seal is a neologism for a mathematical object, that is essentially a subgroup of nimber addition.
The addition of nimbers is the bitwise XOR of non-negative integers. For a finite set ${\displaystyle \{0,...,2^{n}-1\}}$ it forms the Boolean group Z₂ⁿ.

A seal shall be defined as a Boolean function whose family matrix is also the matrix of an equivalence relation.
This implies, that the Boolean function is odd (i.e. that the first entry of it's truth table is true), and that it is the unique odd function in its family.

A seal is also a periodic set partition. The members of its family shall be called its blocks.

seal 1001 0000 0110 0000   (Ж 4471)
This seal has adicity 4, and is shown with (the lowest possible) arity 4, i.e. with a truth table of length 16. The 16×16 matrix on the left is its family matrix. It also describes a partition of the set ${\displaystyle \{0\ldots 15\}}$ into four blocks, which are shown in different colors. On the right the set partition is shown as a vertex coloring of the tesseract graph.   (The 4×4 matrix shows essentially the same.)

The weight of a Boolean function is be the quotient of the sum and the length of its truth table.
The weight of a seal is ${\displaystyle {\frac {1}{2^{d}}}}$, where ${\displaystyle d}$ is its depth.
The unique seal with depth 0 is the tautology. The seals with depth 1 are the negated variadic XORs with one or more arguments.

negated binary Walsh matrix
The seals with depth 1 are the positive rows of a negated binary Walsh matrix.   (The tautology with depth 0 is in the top row.)

The fixed points of Walsh permutations are seals.
The symmetry of Boolean functions is related to seals.

Each seal belongs to a faction, which shall be called its house. A house can also be seen as a clan of seal blocks.
So a house can the represented by the smallest Zhegalkin index of the faction or the clan.   (That of the faction is easier to calculate.)

The seals with arity a form a symmetric Hasse diagram, whose top node is the tautology.
From top to bottom the layers are depth = 0...a. From bottom to top they are rank = 0...a. The number of seals in layer n is ${\displaystyle \mathrm {Oak} (a,n)}$.
For the given arity each seal has a Walsh spectrum. It's non-zero entries are ${\displaystyle 2^{rank}}$ (the weight of the seal's finite truth table).
Their pattern describes another seal in the opposite layer of the Hasse diagram, which shall be called its antipode.
When a seal in house x has an antipode in house y, then all seals in x have antipodes in y. Each house has an antipode for a given arity.

The following seal is the antipode of the example shown above. Their depth (and rank) is 2, and both are on the middle layer of the Hasse diagram.

seal 1000 1000 0001 0001   (Ж 1807)
The Walsh spectrum for arity 4 is (4, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0). Its non-zero entries form the pattern 1001 0000 0110 0000 of the antipode Ж 4471.

1001 0000 0000 1001 (Ж 1911) has cohort {0, 3, 12, 15}. So for arity 4 it is its own antipode. (See this file.)

While a seal has different antipodes for each arity, they all start with the same binary pattern, and then continue with zeros.
In other words, all antipodes of a seal can be described as the same finite set of integers (which corresponds to an entry of sequence Rose).
This set of integers shall be called cohort. Every cohort belongs to a legion. Cohort and legion are properties of Boolean functions in general – not just of seals.

The cardinalities of cohort and legion are a powers of two.
The exponent for the cohort is depth. The exponent for the legion shall be called gravity.
Gravity is similar to valency. The pyramids TwistedLiana and Lonicera have it as one of their dimensions.

truth tables and Zhegalkin indices   (Hasse diagrams of Venn diagrams)
For both arities the green integers on the right correspond to the same seal on the left. E.g. the green 1 corresponds to the tautology, the green 7 to ${\displaystyle \neg (A\oplus B)}$, and the green 15 to ${\displaystyle \neg (A\lor B)}$. truth tables   (entries of Rose) Zhegalkin indices   (entries of Tulip) arity 2 arity 3

sequences Rose and Tulip

Like Boolean functions in general, seals should be seen as periodic truth tables (corresponding to fractions). For a given arity they can be seen as finite truth tables (corresponding to integers). But this approach shall be avoided here. The integer values of the finite truth tables form the sequence Rose = 1, 3, 5, 9, 15... (A190939)     An illustration can be seen here. sequence Rose for arity 4 (adicities 0...4) 1, 3, 5, 9, 15, 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, 257, 513, 771, 1025, 1285, 1545, 2049, 2313, 2565, 3075, 3855, 4097, 4369, 4641, 5185, 6273, 8193, 8481, 8721, 9345, 10305, 12291, 13107, 15555, 16385, 16705, 17025, 17425, 18465, 20485, 21845, 23205, 24585, 26265, 26985, 32769, 33153, 33345, 33825, 34833, 36873, 38505, 39321, 40965, 42405, 43605, 49155, 50115, 52275, 61455, 65535 Hasse diagrams for arities 2 and 3 The gray numbers are the keys to sequence Rose. This is a consistent numbering only for a given arity, but not across arities. Seals with the same label are not equal. E.g. label 0 stands for ${\displaystyle \neg A\land \neg B}$ on the left, but for ${\displaystyle \neg A\land \neg B\land \neg C}$ on the right. Equal seals do not have the same label. E.g. the tautology is labeled 4 on the left, but 15 on the right. Python program to calculate the integer sequence This code uses the Python library discrete helpers. from discretehelpers.a import make_linear_binv arity = 4 depth_to_seals = {depth: None for depth in range(arity)} depth_to_seals[0] = [make_linear_binv(walsh=0, parity=1, arity=arity)] # tautology depth_to_seals[1] = [make_linear_binv(walsh=i, parity=1, arity=arity) for i in range(1, 2 ** arity)] # negated Walsh functions for new_depth in range(2, arity+1): new_weight = 2 ** (arity - new_depth) old_depth = new_depth - 1 old_seals = depth_to_seals[old_depth] new_seals = set() for key_a, seal_a in enumerate(old_seals): for key_b in range(key_a + 1, len(old_seals)): seal_b = old_seals[key_b] candidate = seal_a & seal_b if candidate.weight == new_weight: new_seals.add(candidate) depth_to_seals[new_depth] = sorted(new_seals) lengths = [] sequence = [] for depth, seals in depth_to_seals.items(): rank = arity - depth seal_integers = sorted([seal.intval for seal in seals]) print(f'{rank}: {seal_integers},') lengths.append(len(seal_integers)) sequence += seal_integers print('\n', lengths) print('\n', sorted(sequence)) resulting dictionary rank_to_seals = { 4: [65535], 3: [255, 3855, 13107, 15555, 21845, 23205, 26265, 26985, 38505, 39321, 42405, 43605, 50115, 52275, 61455], 2: [15, 51, 85, 105, 153, 165, 195, 771, 1285, 1545, 2313, 2565, 3075, 4369, 4641, 5185, 6273, 8481, 8721, 9345, 10305, 12291, 16705, 17025, 17425, 18465, 20485, 24585, 33153, 33345, 33825, 34833, 36873, 40965, 49155], 1: [3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769], 0: [1] } A better way to represent the seals by integers is with Zhegalkin indices. They form the sequence Tulip = 1, 3, 5, 7, 15... sequence Tulip for arity 4 (adicities 0...4) 1, 3, 5, 7, 15, 17, 19, 21, 23, 51, 63, 85, 95, 119, 127, 255, 257, 259, 261, 263, 273, 275, 277, 279, 771, 783, 819, 831, 1285, 1295, 1365, 1375, 1799, 1807, 1911, 1919, 3855, 4095, 4369, 4403, 4437, 4471, 4883, 4915, 4959, 4991, 5397, 5439, 5461, 5503, 5911, 5951, 5983, 6007, 13107, 13311, 16191, 16383, 21845, 22015, 24415, 24575, 30583, 30719, 32639, 32767, 65535 illustration:   Zhegalkin indices and truth tables for arity 4 The rows of the green matrix are the twins of those in the red matrix. The latter are the periodic truth tables of the seals. The green integers are their Zhegalkin indices. The tables are divided into adicities 0...4, corresponding to the period length (which is indicated by the grayed-out area). The depth is indicated by the numbers between the two tables.

        0    1     2      3      4     5    6    7
                                                            
0       1                                                     1
1       1    1                                                2
2       1    3     1                                          5
3       1    7     7      1                                  16
4       1   15    35     15      1                           67
5       1   31   155    155     31     1                    374
6       1   63   651   1395    651    63    1              2825
7       1  127  2667  11811  11811  2667  127    1        29212

        0   1   2   3    4    5    6    7

0       1                                         1
1       1   1                                     2
2       1   2   1                                 4
3       1   3   3   1                             8
4       1   4   6   4    1                       16
5       1   5  10  10    5    1                  32
6       1   6  16  22   16    6    1             68
7       1   7  23  43   43   23    7    1       148

triangle Birch with row sums Aster
These sequences count seals whose adicity and valency are equal. ${\displaystyle \mathrm {Aster} (n)}$ is the number of seals whose adicity and valency is n. Among them ${\displaystyle \mathrm {Birch} (n,d)}$ are those with depth d.
triangle Birch (~A139382)    row sums Aster (A135922) d n 0 1 2 3 4 5 6 7 sums 0 1 1 1 0 1 1 2 0 1 1 2 3 0 1 4 1 6 4 0 1 13 11 1 26 5 0 1 40 90 26 1 158 6 0 1 121 670 480 57 1 1330 7 0 1 364 4811 7870 2247 120 1 15414	Birch as a side of Liana and Ivy

illustrated examples:       Oak, Maple (seals with arity 5)       MapleMinor, Sycamore (blocks with adicity 4)

OakDrop(5, 0)=0 OakDrop(5, 1)=ExOak(4, 4)=1 ⋅ 6=16 OakDrop(5, 2)=ExOak(4, 3)=15 ⋅ 8=120 OakDrop(5, 3)=ExOak(4, 2)=35 ⋅ 4=140 OakDrop(5, 4)=ExOak(4, 1)=15 ⋅ 2=30 OakDrop(5, 5)=ExOak(4, 0)=1 ⋅ 1=1 triangles These are the same triangles as shown in the boxes above. They refer to the seals with arity 5. The main content of each image is a refinement of the shown slice of pyramid Ivy. Each column for a valency is refined into columns for each equivalence class. Shown above them are a slice of pyramid Liana and its refinement. Their entries are the column sums of the triangles below. Triangle OakDrop is still called Maple in these images. depth 1 depth 2 depth 3 depth 4

See pyramid Liana. These pyramid pairs are refinements of Oak and Maple, adding axes for valency and gravity.

Hyperpyramids by a, d, v, g