§ 1. Like many mathematical articles, this one begins with a set, specifically S₅ = { 0, 1, 2, 3, 4 }. Variables representing elements of this set are written a, b, c et cetera. Appearing also in the discussion are integers i, j, k, m and n, which are not necessarily elements of S₅. Any arithmetic involving the members of S₅ is done in modulo 5.

The principal undertaking is to introduce 240 ("dozen score" = 12 × 20) binary operations over this set, the operations satisfying two cancellation properties that can be explained after some notation is established. Each operation is denoted by a nonnegative integer followed by the nabla symbol: 0∇, 1∇, 2∇ … 239∇. Variables representing operations are written Z, Y, X et cetera. For an example of such an operation, here is the table that defines 58∇:

58∇		second input
		0	1	2	3	4
first input	0	1	0	4	3	2
	1	3	2	1	0	4
	2	0	4	3	2	1
	3	2	1	0	4	3
	4	4	3	2	1	0

First-order cancellativity (FOC) is the well-known property where:

• If Z (a, b) equals Z (a, c), then b must equal c.
• If Z (a, c) equals Z (b, c), then a must equal b.

• If Z (a + n, b) equals Z (a, b), then n must equal 0.
• If Z (a, b + n) equals Z (a, b), then n must equal 0.

Second-order cancellativity (SOC) is achieved when:

• If Z (a + n, b + n) equals Z (a, b), then n must equal 0.
• If Z (a + n, b − n) equals Z (a, b), then n must equal 0.

There are 240 binary operations on S₅ that satisfy both FOC and SOC, and they are the target of this report. They are all listed in a large table, and formatted as a C-language array.

This report generally uses prefix notation for applications of the operations, but infix notation is equally valid and sometimes clearer:

Z (a, b) ⇔ a Z b

Parentheses and the comma are retained in the prefix notation to reduce confusion, because the arguments are sometimes expressions involving more than one symbol.

§ 2. The nabla numbers of the dozen-scores are based on an obvious manner of lexicographic ordering. The table below is explicit about how the properties of operations m∇ and n∇ are used to establish a relation between integers m and n.

dozen-score lexicographic ordering
first criterion	• If m∇(0, 0) < n∇(0, 0), then m < n. • If m∇(0, 0) > n∇(0, 0), then m > n. • If m∇(0, 0) = n∇(0, 0), then use the second criterion.
second criterion	• If m∇(0, 1) < n∇(0, 1), then m < n. • If m∇(0, 1) > n∇(0, 1), then m > n. • If m∇(0, 1) = n∇(0, 1), then use the third criterion.
third criterion	• If m∇(0, 2) < n∇(0, 2), then m < n. • If m∇(0, 2) > n∇(0, 2), then m > n. • If m∇(0, 2) = n∇(0, 2), then use the fourth criterion.
fourth criterion	• If m∇(0, 3) < n∇(0, 3), then m < n. • If m∇(0, 3) > n∇(0, 3), then m > n. • If m∇(0, 3) = n∇(0, 3), then use the fifth criterion.
fifth criterion	• If m∇(0, 4) < n∇(0, 4), then m < n. • If m∇(0, 4) > n∇(0, 4), then m > n. • If m∇(0, 4) = n∇(0, 4), then use the sixth criterion.
sixth criterion	• If m∇(0, 5) < n∇(0, 5), then m < n. • If m∇(0, 5) > n∇(0, 5), then m > n. • If m∇(0, 5) = n∇(0, 5), then m = n.

§ 3. The dozen-scores sometimes do, and sometimes do not, exhibit familiar algebraic properties.

Because of SOC, the operations are strictly non-commutative: if Z (a, b) equals Z (b, a), then a must equal b.

The standard associative property always fails, but some multiple-operation versions succeed. Also a distributive law works for some combinations of operations. These are discussed in section 6 below.

No operations have a two-sided identity element, although some have a left identity, which means that there exists an a such that Z (a, b) = b for all b. Some others have the corresponding right identity. FOC guarantees that when such a one-sided identity exists, it is unique. Examples:

193∇ second input 0 1 2 3 4 first input 0 4 0 1 2 3 1 2 3 4 0 1 2 0 1 2 3 4 3 3 4 0 1 2 4 1 2 3 4 0	151∇ second input 0 1 2 3 4 first input 0 3 0 2 4 1 1 4 1 3 0 2 2 0 2 4 1 3 3 1 3 0 2 4 4 2 4 1 3 0
left identity = 2	right identity = 1

Only two operations are idempotent in that Z (a, a) = a:

46∇ second input 0 1 2 3 4 first input 0 0 4 3 2 1 1 2 1 0 4 3 2 4 3 2 1 0 3 1 0 4 3 2 4 3 2 1 0 4

21∇ second input 0 1 2 3 4 first input 0 0 2 4 1 3 1 4 1 3 0 2 2 3 0 2 4 1 3 2 4 1 3 0 4 1 3 0 2 4

§ 4. One operation can be systematically altered into other. If X (a, b) = W (a, b) for all a and b, the identity shorthand X ≡ W is often used.

§ 4a. Three self-inverse transformations:

• If Y (a, b) equals Z (b, a), then each is the transpose of the other. By superscript, Y ≡ Z^T
• If Y (a, b) equals Z (a, 4 − b), then each is the vertical reverse of the other. By superscript, Y ≡ Z^V
• If Y (a, b) equals Z (4 − a, b), then each is the horizontal reverse of the other. By superscript, Y ≡ Z^H

100∇ second input 0 1 2 3 4 first input 0 2 0 3 1 4 1 1 4 2 0 3 2 0 3 1 4 2 3 4 2 0 3 1 4 3 1 4 2 0	110∇ second input 0 1 2 3 4 first input 0 2 1 0 4 3 1 0 4 3 2 1 2 3 2 1 0 4 3 1 0 4 3 2 4 4 3 2 1 0	167∇ second input 0 1 2 3 4 first input 0 3 1 4 2 0 1 4 2 0 3 1 2 0 3 1 4 2 3 1 4 2 0 3 4 2 0 3 1 4	213∇ second input 0 1 2 3 4 first input 0 4 1 3 0 2 1 3 0 2 4 1 2 2 4 1 3 0 3 1 3 0 2 4 4 0 2 4 1 3
100∇	110∇ ≡ 100∇T	167∇ ≡ 100∇V	213∇ ≡ 100∇H

The elements unmoved in transposition lie on the principal diagonal.

Successive superscripts are evaluated in the natural fashion: Z^HTV ≡ ((Z^H)^T)^V.

Two notable relations are Z (b, a) = Z^T (a, b) and Z^TH ≡ Z^VT.

The vertical and horizontal reversing operations suggest establishing a unary reversing operation, superscripted with I (for inverse) to avoid confusion with an operation introduced in the next section:

a	0	1	2	3	4
aI = 4 − a	4	3	2	1	0

Thus Z (a, b) = Z^V (a^I, b) = Z^H (a, b^I) = Z^HV (a^I, b^I) = Z^VH (a^I, b^I).

§ 4b. Two cyclical transformations:

• Under row rotation, Y (a, b) equals Z (a + n, b). By superscript, Y ≡ Z^R+n
• Under column rotation, Y (a, b) equals Z (a, b + n). By superscript, Y ≡ Z^C+n

62∇ second input 0 1 2 3 4 first input 0 1 2 0 4 3 1 0 4 3 1 2 2 3 1 2 0 4 3 2 0 4 3 1 4 4 3 1 2 0	107∇ second input 0 1 2 3 4 first input 0 2 0 4 3 1 1 4 3 1 2 0 2 1 2 0 4 3 3 0 4 3 1 2 4 3 1 2 0 4	161∇ second input 0 1 2 3 4 first input 0 3 1 2 0 4 1 2 0 4 3 1 2 4 3 1 2 0 3 1 2 0 4 3 4 0 4 3 1 2
62∇	107∇ ≡ 62∇R+3	161∇ ≡ 62∇C+4

These four statements are equivalent:

• 107∇ ≡ 62∇^R+3
• 107∇^R−3 ≡ 62∇
• 107∇ ≡ 62∇^R−2
• 107∇^R+2 ≡ 62∇

Row and column rotations are not independent:

• (Z^R+n)^T ≡ (Z^T)^C+n
• For half of the operations, Z (a, b) = Z (a + 1, b + 2). For these, Z^R+2 ≡ Z^C+1
• For the other half, Z (a, b) = Z (a + 2, b + 1). For these, Z^R+1 ≡ Z^C+2

There is not an transformation to in general exchange two rows, or two colums, as this could disturb SOC.

§ 4c. Elements of the table can be pairwise swapped, as denoted by a superscript S and two numbers from { 0, 1, 2, 3, 4 }. Only the output values, not the input values, are changed. The highlighted squares of this example show the effect of swapping:

1∇ second input 0 1 2 3 4 first input 0 0 1 2 3 4 1 3 4 0 1 2 2 1 2 3 4 0 3 4 0 1 2 3 4 2 3 4 0 1	43∇ second input 0 1 2 3 4 first input 0 0 4 2 3 1 1 3 1 0 4 2 2 4 2 3 1 0 3 1 0 4 2 3 4 2 3 1 0 4
1∇ ≡ 43∇S1,4	43∇ ≡ 1∇S1,4

The swap being pairwise, Z^Sm,n ≡ Z^Sn,m. If m = n, nothing happens.

Canonization (or canonicalization) is the process of performing a sequence of pairwise swaps ultimately producing an operation where Z(0, a) = a; this condition is equivalent to saying that 0 is a left identity of Z. Such a canonical form helps to reveal the "shape" of the operation. Anywhere between zero and four swaps will be necessary. Example:

159∇ second input 0 1 2 3 4 first input 0 3 1 0 4 2 1 4 2 3 1 0 2 1 0 4 2 3 3 2 3 1 0 4 4 0 4 2 3 1	7∇ second input 0 1 2 3 4 first input 0 0 1 3 4 2 1 4 2 0 1 3 2 1 3 4 2 0 3 2 0 1 3 4 4 3 4 2 0 1	3∇ second input 0 1 2 3 4 first input 0 0 1 2 4 3 1 4 3 0 1 2 2 1 2 4 3 0 3 3 0 1 2 4 4 2 4 3 0 1	1∇ second input 0 1 2 3 4 first input 0 0 1 2 3 4 1 3 4 0 1 2 2 1 2 3 4 0 3 4 0 1 2 3 4 2 3 4 0 1
159∇	7∇ ≡ 159∇S0,3	3∇ ≡ 7∇S3,2	1∇ ≡ 3∇S4,3

For the dozen-scores:

• Transposition, vertical reversal, or horizontal reversal does change the operation's canonical form.
• Rotation by row, rotation by column, or element swapping does not change the operation's canonical form.

If Z reduces to 0∇ and Y reduces to 1∇, then Z and Y taken together form a Graeco-Latin square. Each cell of the table contains a different ordered pair. Example:

60∇, 104∇		second input
		0	1	2	3	4
first input	0	1, 2	2, 0	0, 4	3, 1	4, 3
	1	0, 1	3, 3	4, 2	1, 0	2, 4
	2	4, 0	1, 4	2, 1	0, 3	3, 2
	3	2, 3	0, 2	3, 0	4, 4	1, 1
	4	3, 4	4, 1	1, 3	2, 2	0, 0

§ 5. The dozen-scores can be partitioned into six subsets (quadragintas) of 40 operations each, as shown in the tables below.

• Within each column of a quadraginta (for instance 0∇, 110∇, 167∇, 213∇, 181∇, 238∇, 20∇, 100∇), the operations are related by transposition, vertical reversion, and horizontal reversion.
• Within each row of a quadraginta (for instance 0∇, 192∇, 180∇, 129∇, 66∇), the operations are related by row rotation or column rotation.
• Within the first four rows of each quadraginta (red), Z (0, 0) = Z (2, 1); within the last four (green), Z (0, 0) = Z (1, 2).
• Element swapping carries an operation from one quadraginta to another. For instance, 0∇ is in quadraginta A, but 48∇ ≡ 0∇^S0,1 is in quadraginta F.

quadraginta A 0∇ 192∇ 180∇ 129∇ 66∇ 110∇ 59∇ 47∇ 239∇ 173∇ 167∇ 101∇ 89∇ 26∇ 218∇ 213∇ 150∇ 138∇ 72∇ 21∇ 181∇ 128∇ 67∇ 1∇ 193∇ 238∇ 172∇ 111∇ 58∇ 46∇ 20∇ 212∇ 151∇ 139∇ 73∇ 100∇ 88∇ 27∇ 219∇ 166∇	quadraginta B 2∇ 144∇ 228∇ 141∇ 71∇ 108∇ 54∇ 35∇ 191∇ 221∇ 215∇ 105∇ 77∇ 38∇ 170∇ 165∇ 198∇ 126∇ 84∇ 17∇ 229∇ 140∇ 70∇ 3∇ 145∇ 190∇ 220∇ 109∇ 55∇ 34∇ 16∇ 164∇ 199∇ 127∇ 85∇ 104∇ 76∇ 39∇ 171∇ 214∇	quadraginta C 5∇ 194∇ 132∇ 177∇ 78∇ 158∇ 57∇ 42∇ 227∇ 125∇ 119∇ 149∇ 93∇ 14∇ 230∇ 209∇ 102∇ 186∇ 60∇ 33∇ 133∇ 176∇ 79∇ 4∇ 195∇ 226∇ 124∇ 159∇ 56∇ 43∇ 32∇ 208∇ 103∇ 187∇ 61∇ 148∇ 92∇ 15∇ 231∇ 118∇
quadraginta D 6∇ 96∇ 216∇ 189∇ 83∇ 156∇ 50∇ 23∇ 143∇ 233∇ 210∇ 153∇ 65∇ 36∇ 122∇ 117∇ 203∇ 174∇ 86∇ 29∇ 217∇ 188∇ 82∇ 7∇ 97∇ 142∇ 232∇ 157∇ 51∇ 22∇ 28∇ 116∇ 202∇ 175∇ 87∇ 152∇ 64∇ 37∇ 123∇ 211∇	quadraginta E 9∇ 146∇ 120∇ 225∇ 90∇ 206∇ 53∇ 30∇ 179∇ 137∇ 114∇ 197∇ 81∇ 12∇ 182∇ 161∇ 107∇ 234∇ 62∇ 45∇ 121∇ 224∇ 91∇ 8∇ 147∇ 178∇ 136∇ 207∇ 52∇ 31∇ 44∇ 160∇ 106∇ 235∇ 63∇ 196∇ 80∇ 13∇ 183∇ 115∇	quadraginta F 11∇ 98∇ 168∇ 237∇ 95∇ 204∇ 48∇ 18∇ 131∇ 185∇ 162∇ 201∇ 69∇ 24∇ 134∇ 113∇ 155∇ 222∇ 74∇ 41∇ 169∇ 236∇ 94∇ 10∇ 99∇ 130∇ 184∇ 205∇ 49∇ 19∇ 40∇ 112∇ 154∇ 223∇ 75∇ 200∇ 68∇ 25∇ 135∇ 163∇

The following formulas show more specifically how the dozen-scores within a column or row of a quadraginta table are connected, by giving as examples some relations within the first column and the first row of quadraginta B:

2∇ ≡	16∇T ≡	229∇V ≡	190∇H
108∇ ≡	104∇T ≡	190∇V ≡	229∇H
215∇ ≡	229∇T ≡	104∇V ≡	16∇H
165∇ ≡	190∇T ≡	16∇V ≡	104∇H
229∇ ≡	215∇T ≡	2∇V ≡	108∇H
190∇ ≡	165∇T ≡	108∇V ≡	2∇H
16∇ ≡	2∇T ≡	165∇V ≡	215∇H
104∇ ≡	108∇T ≡	215∇V ≡	165∇H

2∇ ≡ 144∇^C+1 ≡ 228∇^C+2 ≡ 141∇^C+3 ≡ 71∇^C+4

The operations in quadraginta A exhibit linearity in the following sense: for each operation there exist i, j and k such that Z(a, b) = i × a + j × b + k. With 128∇ for instance, i = 3, j = 1, and k = 2:

128∇		second input
		0	1	2	3	4
first input	0	2	3	4	0	1
	1	0	1	2	3	4
	2	3	4	0	1	2
	3	1	2	3	4	0
	4	4	0	1	2	3

For a linear operation Z, the coëfficients are easily found:

• i = Z(1, 0) − Z(0, 0)
• j = Z(0, 1) − Z(0, 0)
• k = Z(0, 0)

With linear operations, four relations can never occur, because they would be connected with a violation of FOC or SOC:

• i = 0
• j = 0
• i + j = 0
• i − j = 0

In the next section are more examples of how quadraginta A is distinguished from the others.

§ 6. Combining operations.

§ 6a. There are 48,000 combinations (out of 240⁴ possibilities) of operations that satisfy this four-operation generalization of the associative law, here written in both prefix and infix notations:

Z (a, Y (b, c)) = X (W (a, b), c)
a Z (b Y c) = (a W b) X c

Necessary conditions are that operations Z and X must come from the same quadraginta, and operations Y and W must come from quadraginta A. Never does the particular case Z ≡ Y ≡ X ≡ W succeed.

§ 6b. There are 576,000 combinations (of 240⁴) of operations that satisfy this next version of associativity:

Z (Y (a, b), c) = X (W (a, b), c)
(a Y b) Z c = (a W b) X c

A necessary condition is that Z and X must come from the same quadraginta. Always successful is Z ≡ Y ≡ X ≡ W.

§ 6c. There are 160 combinations (of 240²) of operations, all from quadraginta A, that satisfy this distributive law:

Z (Y (a, b), c) = Y (Z (a, c), Z (b, c))
(a Y b) Z c = (a Z c) Y (b Z c)

Only 21∇ and 46∇, the idempotent operations, fulfill the constraint Y ≡ Z.

§ 6d. A general approach to composite operations is lengthy enough to need its own page.

§ 7. Thus far, great attention has been paid to a collection of 240 operations over the set S₅ = { 0, 1, 2, 3, 4 }. Why not the more general S_n = { 0, 1, 2 … n − 1 }? There is an important result about binary operations over S_n that are first- and second-order cancellative:

• They do not exist when n is a multiple of 2 or 3.
• They do exist when n is not a multiple of 2 or 3.

For each of S₇ and S₁₁, there are two substantively different configurations. Examples, in canonical form, appear below preceded by 1∇ for comparison:

1∇ second input 0 1 2 3 4 first input 0 0 1 2 3 4 1 3 4 0 1 2 2 1 2 3 4 0 3 4 0 1 2 3 4 2 3 4 0 1

1∇(a, b) = 1∇(a + 1, b + 2)

U1 second input 0 1 2 3 4 5 6 first input 0 0 1 2 3 4 5 6 1 5 6 0 1 2 3 4 2 3 4 5 6 0 1 2 3 1 2 3 4 5 6 0 4 6 0 1 2 3 4 5 5 4 5 6 0 1 2 3 6 2 3 4 5 6 0 1	U2 second input 0 1 2 3 4 5 6 first input 0 0 1 2 3 4 5 6 1 4 5 6 0 1 2 3 2 1 2 3 4 5 6 0 3 5 6 0 1 2 3 4 4 2 3 4 5 6 0 1 5 6 0 1 2 3 4 5 6 3 4 5 6 0 1 2
U1(a, b) = U1(a + 1, b + 2)	U2(a, b) = U2(a + 1, b + 3)

U3 second input 0 1 2 3 4 5 6 7 8 9 10 first input 0 0 1 2 3 4 5 6 7 8 9 10 1 9 10 0 1 2 3 4 5 6 7 8 2 7 8 9 10 0 1 2 3 4 5 6 3 5 6 7 8 9 10 0 1 2 3 4 4 3 4 5 6 7 8 9 10 0 1 2 5 1 2 3 4 5 6 7 8 9 10 0 6 10 0 1 2 3 4 5 6 7 8 9 7 8 9 10 0 1 2 3 4 5 6 7 8 6 7 8 9 10 0 1 2 3 4 5 9 4 5 6 7 8 9 10 0 1 2 3 10 2 3 4 5 6 7 8 9 10 0 1	U4 second input 0 1 2 3 4 5 6 7 8 9 10 first input 0 0 1 2 3 4 5 6 7 8 9 10 1 8 9 10 0 1 2 3 4 5 6 7 2 5 6 7 8 9 10 0 1 2 3 4 3 2 3 4 5 6 7 8 9 10 0 1 4 10 0 1 2 3 4 5 6 7 8 9 5 7 8 9 10 0 1 2 3 4 5 6 6 4 5 6 7 8 9 10 0 1 2 3 7 1 2 3 4 5 6 7 8 9 10 0 8 9 10 0 1 2 3 4 5 6 7 8 9 6 7 8 9 10 0 1 2 3 4 5 10 3 4 5 6 7 8 9 10 0 1 2
U3(a, b) = U3(a + 1, b + 2)	U4(a, b) = U4(a + 1, b + 3)

All the other FOC and SOC operations over S₇ and S₁₁ can be found by making the obvious adaptations to the following transformations that were defined for S₅:

• Z^T
• Z^V
• Z^H
• Z^R+n
• Z^C+n
• Z^Sm,n

When n ≤ 11, each operation over S_n can, through canonization, be transformed into a linear function (section 5). However, this property no longer applies when n ≥ 13, the point at which the operations become far more complicated. The following is an operation over S₁₃ that, although it is in canonical form, is conspicuously non-linear:

U5 second input 0 1 2 3 4 5 6 7 8 9 10 11 12 first input 0 0 1 2 3 4 5 6 7 8 9 10 11 12 1 9 10 11 12 0 1 2 3 4 5 6 7 8 2 5 6 7 8 9 10 11 12 0 1 2 3 4 3 12 0 1 2 3 4 5 6 7 8 9 10 11 4 2 3 4 5 6 7 8 9 10 11 12 0 1 5 7 8 9 10 11 12 0 1 2 3 4 5 6 6 10 11 12 0 1 2 3 4 5 6 7 8 9 7 4 5 6 7 8 9 10 11 12 0 1 2 3 8 1 2 3 4 5 6 7 8 9 10 11 12 0 9 8 9 10 11 12 0 1 2 3 4 5 6 7 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 11 12 0 1 2 12 6 7 8 9 10 11 12 0 1 2 3 4 5

U5(a, b) = U5(a + m, b + n) cannot be satisfied

§ 8. The three-operand case can also be glimpsed. The following definitions are practically inevitable, and assume modulo-n arithmetic when the operation is over S_n:

FOC is satisfied when:

• If Z (a + n, b, c) = Z (a, b, c), then n = 0.
• If Z (a, b + n, c) = Z (a, b, c), then n = 0.
• If Z (a, b, c + n) = Z (a, b, c), then n = 0.

SOC is satisfied when:

• If Z (a + n, b + n, c) = Z (a, b, c), then n = 0.
• If Z (a + n, b − n, c) = Z (a, b, c), then n = 0.
• If Z (a, b + n, c + n) = Z (a, b, c), then n = 0.
• If Z (a, b + n, c − n) = Z (a, b, c), then n = 0.
• If Z (a + n, b, c + n) = Z (a, b, c), then n = 0.
• If Z (a − n, b, c + n) = Z (a, b, c), then n = 0.

Third-order cancellativity (TOC) is satisfied when:

• If Z (a + n, b + n, c + n) = Z (a, b, c), then n = 0.
• If Z (a + n, b + n, c − n) = Z (a, b, c), then n = 0.
• If Z (a + n, b − n, c + n) = Z (a, b, c), then n = 0.
• If Z (a + n, b − n, c − n) = Z (a, b, c), then n = 0.

Within the formulas there are more plus signs than minus signs, but this is not a problem because n itself can be positive or negative,.

SOC and TOC form two different kinds of pandiagonality. (Analogous are the face-centered and body-centered cubic lattices.)

Some existences:

• With S₅:
  • There are 240 two-operand operations that satisfy FOC and SOC, as discussed above.
  • There are no three-operand operations that satisfy FOC and SOC.
• With S₇:
  • There are three-operand operations that satisfy FOC and SOC.
  • There are no three-operand operations that satisfy FOC, SOC and TOC.
• With S₁₁:
  • There are three-operand operations that satisfy FOC, SOC and TOC, all of them linear. An example is U₆(a, b, c) = 5 × a + 8 × b + 1 × c + 0.