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 it forms the Boolean groupZ2n.
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 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 , where 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.)
Equivalence classes (EC) of seals are factions. Those of blocks are clans.
Thus an EC 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 .
For the given arity each seal has a Walsh spectrum. It's non-zero entries are (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 EC x has an antipode in EC y, then all seals in x have antipodes in y.
Each EC has an antipode for a given arity. (Some EC are their own antipodes.)
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 with the same finite set of integers. This set corresponds to an entry of sequence Rose.
(So the entries of Rose do indeed correspond to seals, and not just to their finite truth tables.)
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.
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.
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.
Arityn is short for adicity ≤ n.
Sequences, triangles and pyramids using arity are marked with a wave 🌊, those using adicity are marked with a drop 💧.
🌊 entries are sums of 💧 columns.
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seals and EC by arity and depth: symmetric triangles Oak and Elm
The triangle Oak (left) shows the number of seals by arity (rows) and depth (columns). These are 2-binomial coefficients. Arity n is short for adicity ≤ n.
The triangle Elm (right) shows the corresponding numbers of equivalence classes. (Not to be confused with Pascal's triangle.)
These sequences count seals whose adicity and valency are equal. is the number of seals whose adicity and valency is n.
Among them are those with depth d.
illustrated examples: Oak, Maple (seals with arity 5) MapleMinor, Sycamore (blocks with adicity 4)
Maple(5, 0)=0
Maple(5, 1)=MapleMinor(4, 4)=1 ⋅ 6=16
Maple(5, 2)=MapleMinor(4, 3)=15 ⋅ 8=120
Maple(5, 3)=MapleMinor(4, 2)=35 ⋅ 4=140
Maple(5, 4)=MapleMinor(4, 1)=15 ⋅ 2=30
Maple(5, 5)=MapleMinor(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.
depth 1
depth 2
depth 3
depth 4
🌊 pyramid Liana
overview
arity: top to bottom depth: back to front valency: left to right
Indices in the image go from 1 to 7. Liana is always 1 where depth and valency are 0. But this column is not shown in the images.
The sum ignoring valency is triangle Oak.
The sum ignoring depth is triangle Ash.
The layer sums (and row sums of these triangles) are sequence Daisy (A006116).
fixed arity (depth × valency matrices)
The row sums are rows of triangle Oak. The column sums are rows of triangle Ash. The total sums are entries of Daisy.
arity 0
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
3
4
5
6
7
Σ
1
1
arity 1
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
3
4
5
6
7
Σ
1
1
2
arity 2
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
1
3
2
1
1
3
4
5
6
7
Σ
1
2
2
5
arity 3
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
3
3
1
7
2
3
4
7
3
1
1
4
5
6
7
Σ
1
3
6
6
16
arity 4
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
4
6
4
1
15
2
6
16
13
35
3
4
11
15
4
1
1
5
6
7
Σ
1
4
12
24
26
67
arity 5
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
5
10
10
5
1
31
2
10
40
65
40
155
3
10
55
90
155
4
5
26
31
5
1
1
6
7
Σ
1
5
20
60
130
158
374
arity 6
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
6
15
20
15
6
1
63
2
15
80
195
240
121
651
3
20
165
540
670
1395
4
15
156
480
651
5
6
57
63
6
1
1
7
Σ
1
6
30
120
390
948
1330
2825
arity 7
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
7
21
35
35
21
7
1
127
2
21
140
455
840
847
364
2667
3
35
385
1890
4690
4811
11811
4
35
546
3360
7870
11811
5
21
399
2247
2667
6
7
120
127
7
1
1
Σ
1
7
42
210
910
3318
9310
15414
29212
fixed depth (arity × valency matrices)
The row sums are columns of triangle Oak.
depth 0
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
depth 1
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
2
1
3
3
3
3
1
7
4
4
6
4
1
15
5
5
10
10
5
1
31
6
6
15
20
15
6
1
63
7
7
21
35
35
21
7
1
127
depth 2
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
1
1
3
3
4
7
4
6
16
13
35
5
10
40
65
40
155
6
15
80
195
240
121
651
7
21
140
455
840
847
364
2667
depth 3
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
1
1
4
4
11
15
5
10
55
90
155
6
20
165
540
670
1395
7
35
385
1890
4690
4811
11811
depth 4
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
1
1
5
5
26
31
6
15
156
480
651
7
35
546
3360
7870
11811
depth 5
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
1
1
6
6
57
63
7
21
399
2247
2667
depth 6
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
1
1
7
7
120
127
depth 7
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
7
1
1
sum: triangle Ash
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
2
1
2
2
5
3
1
3
6
6
16
4
1
4
12
24
26
67
5
1
5
20
60
130
158
374
6
1
6
30
120
390
948
1330
2825
7
1
7
42
210
910
3318
9310
15414
29212
fixed valency (arity × depth matrices)
The row sums are columns of triangle Ash.
valency 0
d
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
valency 1
d
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
valency 2
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
1
1
2
3
3
3
6
4
6
6
12
5
10
10
20
6
15
15
30
7
21
21
42
valency 3
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
1
4
1
6
4
4
16
4
24
5
10
40
10
60
6
20
80
20
120
7
35
140
35
210
valency 4
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
1
13
11
1
26
5
5
65
55
5
130
6
15
195
165
15
390
7
35
455
385
35
910
valency 5
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
1
40
90
26
1
158
6
6
240
540
156
6
948
7
21
840
1890
546
21
3318
valency 6
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
1
121
670
480
57
1
1330
7
7
847
4690
3360
399
7
9310
valency 7
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
7
1
364
4811
7870
2247
120
1
15414
sum: triangle Oak
d
a
0
1
2
3
4
5
6
7
Σ
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
other sides
💧 pyramid Ivy
overview
adicity: top to bottom depth: back to front valency: left to right
Indices in the image go from 1 to 7.
The entry Ivy(0, 0, 0) = 1 is not shown in the images.
The sum ignoring valency is triangle Maple.
The sum ignoring depth is triangle Aspen.
The layer sums (and row sums of these triangles) are sequence Dahlia (A182176).
The pyramid sides in the back (depth = 1) and front (valency − depth = 0) are Pascal's triangle.
fixed adicity (depth × valency matrices)
The row sums are rows of triangle Maple. The column sums are rows of triangle Aspen. The total sums are entries of Dahlia.