§1. An embee is a square array of nonnegative integers:

• Each element must be less than the embee's modulus, symbolized 𝕄.
• The length of an edge of the array is its breadth, symbolized 𝔹.
• Each of 𝕄 and 𝔹 is a positive integer, with a minimum practical value of 2.
• Under the rules of modular arithmetic, all the numbers in each row, each column, each superior diagonal, and each inferior diagonal, must sum to zero.

The pronunciation of "𝕄𝔹" is what yields the name embee.

Any row, column, superior, or inferior is termed a straight.

The embee Z, all of whose elements are zero, exists for all 𝕄 and 𝔹.

In this report, embees will usually have names like E, F, and G. The modulus of E can be written E_𝕄; the breadth E_𝔹.

It is obvious what the rows and columns of a square array would be, but an example will shed light on what the two kinds of diagonals are. Given embee E, with 𝔹 = 4 and 𝕄 = any value:

E
E00	E01	E02	E03
E10	E11	E12	E13
E20	E21	E22	E23
E30	E31	E32	E33

The four superior diagonals are:

• E₀₀, E₁₁, E₂₂, E₃₃
• E₀₁, E₁₂, E₂₃, E₃₀
• E₀₂, E₁₃, E₂₀, E₃₁
• E₀₃, E₁₀, E₂₁, E₃₂

• E₀₃, E₁₂, E₂₁, E₃₀
• E₀₂, E₁₁, E₂₀, E₃₃
• E₀₁, E₁₀, E₂₃, E₃₂
• E₀₀, E₁₃, E₂₂, E₃₁

The number of superiors is always equal to the breadth, as is the number of inferiors.

Although the names superior and inferior are not particularly descriptive, no better term has been found.

§2. Two embees are equivalent to each other if they have the same breadth and modulus, and if they are related directly or indirectly by any sequence of the equivalence transformations displayed in this section. A notation for equivalence is E ≍ F.

The transformations are best explained with examples, as follows. Given embee E as before:

E E00 E01 E02 E03 E10 E11 E12 E13 E20 E21 E22 E23 E30 E31 E32 E33

There are four mirror operations:

mirror_row (E) E30 E31 E32 E33 E20 E21 E22 E23 E10 E11 E12 E13 E00 E01 E02 E03

mirror_col (E) E03 E02 E01 E00 E13 E12 E11 E10 E23 E22 E21 E20 E33 E32 E31 E30

mirror_sup (E) E00 E10 E20 E30 E01 E11 E21 E31 E02 E12 E22 E32 E03 E13 E23 E33

mirror_inf (E) E33 E23 E13 E03 E32 E22 E12 E02 E31 E21 E11 E01 E30 E20 E10 E00

There is an operation that rotates the embee by a multiple of ninety degrees. The numerical argument can be any integer, which will be interpreted in modulo 4. An argument of 0 gives the identity operation. Shown as examples are +1, +2, and +3:

rotate (E, +1) E30 E20 E10 E00 E31 E21 E11 E01 E32 E22 E12 E02 E33 E23 E13 E03

rotate (E, +2) E33 E32 E31 E30 E23 E22 E21 E20 E13 E12 E11 E10 E03 E02 E01 E00

rotate (E, +3) E03 E13 E23 E33 E02 E12 E22 E32 E01 E11 E21 E31 E00 E10 E20 E30

There are two shift operations, where the numerical argument can be any integer. Shown for example is +1:

shift_rows (E, +1) E30 E31 E32 E33 E00 E01 E02 E03 E10 E11 E12 E13 E20 E21 E22 E23

shift_cols (E, +1) E03 E00 E01 E02 E13 E10 E11 E12 E23 E20 E21 E22 E33 E30 E31 E32

With embees E, F, and G, it is easy to show that:

• E ≍ E
• If E ≍ F, then F ≍ E
• If E ≍ F, and F ≍ G, then E ≍ G

In many cases, there is more than one way to write an equivalence. For instance,

rotate (shift_rows (E, +1), +1) = shift_cols (rotate (E, +1), −1).

If two embees differ in breadth or modulus or both, equivalence is not defined.

§3. Two embees of the same breadth and the same modulus may be lexicographically compared. The manner of applying this to embees is unsurprising, and employs the row-major ordering used for storing two-dimensional arrays in many computer environments. Here is an example with embees E and F:

• if E₀₀ < F₀₀, then E < F;
• else if E₀₀ > F₀₀, then E > F;
• else if E₀₁ < F₀₁, then E < F;
• else if E₀₁ > F₀₁, then E > F;
• … and continuing through the first row.

If there is no difference between the respective first rows, the second rows are compared, and then the third and so forth. If all corresponding elements are equal, then E = F.

If two embees differ in breadth or modulus or both, no equality or inequalies are defined.

§4. The equivalences of §2 combined with the lexicographical ordering of §3 allows us to establish canonical forms for embees. The principle is that if E ≍ F and E < F, then F is NOT in a canonical form. Every embee does have a unique canonical form.

As an example, below is an ordinary embee (𝕄 = 8) in canonical form. The numbers have been subscripted to make it easier to see how the elements move around under equivalence tranformations.

1aw 2ax 2ay 3az 4bw 5bx 3by 4bz 6cw 5cx 7cy 6cz 5dw 4dx 4dy 3dz

Next are four non-canonical embees equivalent to the one above:

4bw 5bx 3by 4bz 1aw 2ax 2ay 3az 5dw 4dx 4dy 3dz 6cw 5cx 7cy 6cz

3dz 4dy 4dx 5dw 3az 2ay 2ax 1aw 4bz 3by 5bx 4bw 6cz 7cy 5cx 6cw

2ax 5bx 5cx 4dx 2ay 3by 7cy 4dy 3az 4bz 6cz 3dz 1aw 4bw 6cw 5dw

4dy 7cy 3by 2ay 4dx 5cx 5bx 2ax 5dw 6cw 4bw 1aw 3dz 6cz 4bz 3az

§5. The census of an embee is simply a listing of how many elements are of each value. If two embees are equivalent, they have the same census. As an example, each of the embees within this section has 2 zeroes, 4, ones, 3 twos, 3 threes, and 4 fours. Hence its census is (2, 4, 3, 3, 4).

Note that two embees with the same census might not be equivalent. Examples are the four embees below, with 𝕄 = 4 and 𝔹 = 4. All have the census (4, 4, 4, 4), and are all canonical forms. Because they are distinct canonical forms, however, they cannot be equivalent.

0 0 0 0 1 3 1 3 2 2 2 2 1 3 1 3

0 0 1 3 0 2 1 1 2 2 3 1 2 0 3 3

0 1 0 3 2 3 2 1 0 1 0 3 2 3 2 1

0 1 2 1 2 3 0 3 0 1 2 1 2 3 0 3

§6. Here are some operations that preserve embeeness, and can be categorized by their preservation (or not) of modulus and breadth. All arithmetic on elements is, of course, modular.

§6A. Output 𝕄 = input 𝕄, and output 𝔹 = input 𝔹:

G = −E is formed by negating every element of E. Of course, −Z = Z.

G = E + F is formed by adding corresponding elements of E and F. As expected, E − F = E + (−F). Z is the identity augend, addend and subtrahend; but not minuend.

If n is any integer, then G = E × n = n × E is formed by multiplying every element of E by n. Here is an example, where G = E × 3: E, with E𝕄 = 8 1 2 2 3 4 5 3 4 6 5 7 6 5 4 4 3 G, with G𝕄 = 8 3 6 6 1 4 7 1 4 2 7 5 2 7 4 4 1

§6B. Output 𝔹 = input 𝔹:

If n is any nonzero integer, then G = E ×× n = n ×× E is formed by multiplying E𝕄 by the absolute value of n to obtain G𝕄, and then multiplying every element of E by n. Here is an example, where G = E ×× 3: E, where E𝕄 = 8 1 2 2 3 4 5 3 4 6 5 7 6 5 4 4 3 G, where G𝕄 = 24 3 6 6 9 12 15 9 12 18 15 21 18 15 12 12 9 With this operation, there is no modular reassessment.

With embee E, if n is an integer that meets three requirements: n is greater than 1; n is less than E𝕄 n is a factor of E𝕄 then G = E ÷ n is formed by dividing E𝕄 by the absolute value of n to form G𝕄, and then reassessing the elements of E by the new modulus. Note that the elements themselves are not divided by anything. Here are examples, with G = E ÷ 2 and H = E ÷ 3: E, where E𝕄 = 6 0 5 4 3 2 5 1 4 2 0 0 4 2 2 1 1 G, where G𝕄 = 3 0 2 1 0 2 2 1 1 2 0 0 1 2 2 1 1 H, where H𝕄 = 2 0 1 0 1 0 1 1 0 0 0 0 0 0 0 1 1

§6C. Output 𝕄 = input 𝕄:

G = repeat (E, n) fills G with copies of E. In the following example, G = repeat (E, 2), and E𝕄 = G𝕄 = 8: E, where E𝔹 = 4 1 2 2 3 4 5 3 4 6 5 7 6 5 4 4 3 G, where G𝔹 = 8 1 2 2 3 1 2 2 3 4 5 3 4 4 5 3 4 6 5 7 6 6 5 7 6 5 4 4 3 5 4 4 3 1 2 2 3 1 2 2 3 4 5 3 4 4 5 3 4 6 5 7 6 6 5 7 6 5 4 4 3 5 4 4 3 No modular reassessment is necessary.

With G = expand (E, n), each element of E appears once in G, separated by n − 1 rows and columns of zeroes. In the following example, G = expand (E, 2), and E𝕄 = G𝕄 = 8: E, where E𝔹 = 4 1 2 2 3 4 5 3 4 6 5 7 6 5 4 4 3 G, where G𝔹 = 8 1 0 2 0 2 0 3 0 0 0 0 0 0 0 0 0 4 0 5 0 3 0 4 0 0 0 0 0 0 0 0 0 6 0 5 0 7 0 6 0 0 0 0 0 0 0 0 0 5 0 4 0 4 0 3 0 0 0 0 0 0 0 0 0 No modular reassessment is necessary.

§7. For selected combinations of breadth and modulus, the table below displays the quantity of all embees, and the the quantity of embees in canonical form. The breadth of four receives an appendix because its quantities are both interesting and manageable to calculate.

embee quantities		𝕄 = 2	𝕄 = 3	𝕄 = 4	𝕄 = 5	𝕄 = 6	𝕄 = 7	𝕄 = 8	𝕄 = 9
𝔹 = 2	all	2	1	2	1	2	1	2	1
	canon	2	1	2	1	2	1	2	1
𝔹 = 3	all	1	27 = 33	1	1	27 = 33	1	1	27 = 33
	canon	1	5	1	1	5	1	1	5
𝔹 = 4	all	128 = 27	81 = 34	4,096 = 212	625 = 54	10,368 = 27·34	2,401 = 74	65,536 = 216	6,561 = 38
	canon	13	6	111	21	181	55	862	120
𝔹 = 5	all	256 = 28	6,561 = 38	65,536 = 216	48,828,125 = 511
	canon	7	57	400	246,210
𝔹 = 6	all	131,072 = 217
	canon	772
𝔹 = 7	all	16,777,216 = 224
	canon	43,624
appendix		𝕄 = 10	𝕄 = 11	𝕄 = 12	𝕄 = 13	𝕄 = 14	𝕄 = 15	𝕄 = 16	𝕄 = 17
𝔹 = 4	all	80,000 = 27·54	14,641 = 114	331,776 = 212·34	28,561 = 134	307,328 = 27·74	50,625 = 34·54	1,048,576 = 220	83,521 = 174
	canon	963	231	3,517	406	3,199	666	10,140	1,035

The numbers came from a computer program that did a brute-force search. The general formula is not yet known, although some patterns are beginning to emerge, particularly when the modulus is a prime number.

§8. All the embees discussed so far are two-dimensional (𝔻 = 2). The investigation can be extended to higher dimensionalities, although that is not attempted in this report.

The number of straights (𝕊) grows very quickly, with the following general formula:

𝕊 = 𝔹^{𝔻 − 1} × (3^𝔻 − 1) ÷ 2

Note that 𝕊 is independent of the modulus. Here is a table with selected values:

𝕊	𝔹 = 2	𝔹 = 3	𝔹 = 4	𝔹 = 5
𝔻 = 2	8	12	16	20
𝔻 = 3	52	117	208	325
𝔻 = 4	320	1,080	2,560	5,000
𝔻 = 5	1,936	9,801	30,976	75,625

An example is worthwhile. For the case 𝔻 = 3, 𝕄 = 4, and 𝔹 = 4, here are the thirteen straights that pass through (0, 0, 0):

• Perpendicular to two edges (parallel to one edge):
  • (0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3)
  • (0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 3, 0)
  • (0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0)
• Perpendicular to one edge (parallel to a face diagonal):
  • (0, 0, 0), (0, 1, 1), (0, 2, 2), (0, 3, 3)
  • (0, 0, 0), (1, 1, 0), (2, 2, 0), (3, 3, 0)
  • (0, 0, 0), (1, 0, 1), (2, 0, 2), (3, 0, 3)
  • (0, 0, 0), (0, 1, 3), (0, 2, 2), (0, 3, 1)
  • (0, 0, 0), (1, 3, 0), (2, 2, 0), (3, 1, 0)
  • (0, 0, 0), (3, 0, 1), (2, 0, 2), (1, 0, 3)
• Perpendicular to zero edges (parallel to a body diagonal:
  • (0, 0, 0), (1, 1, 1), (2, 2, 2), (3, 3, 3)
  • (0, 0, 0), (1, 1, 3), (2, 2, 2), (3, 3, 1)
  • (0, 0, 0), (1, 3, 1), (2, 2, 2), (3, 1, 3)
  • (0, 0, 0), (3, 1, 1), (2, 2, 2), (1, 3, 3)

The criterion of "perpendicular to n edges" generalizes to straights of higher dimensionality, and is useful for organizing them.