§ 1. Like many mathematical articles, this one begins with a set, specifically S_{5} = { 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 S_{5}. Any arithmetic involving the members of S_{5} 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 
Firstorder cancellativity (FOC) is the wellknown property where:
Secondorder cancellativity (SOC) is achieved when:
There are 240 binary operations on S_{5} 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 Clanguage 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 dozenscores 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.
dozenscore 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 dozenscores sometimes do, and sometimes do not, exhibit familiar algebraic properties.
Because of SOC, the operations are strictly noncommutative: if Z (a, b) equals Z (b, a), then a must equal b.
The standard associative property always fails, but some multipleoperation versions succeed. Also a distributive law works for some combinations of operations. These are discussed in section 6 below.
No operations have a twosided 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 onesided identity exists, it is unique. Examples:

 
left identity = 2  right identity = 1 
Only two operations are idempotent in that Z (a, a) = a:


§ 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 selfinverse transformations:



 
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 
a^{I} = 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:


 
62∇  107∇ ≡ 62∇^{R+3}  161∇ ≡ 62∇^{C+4} 
These four statements are equivalent:
Row and column rotations are not independent:
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∇ ≡ 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∇  7∇ ≡ 159∇^{S0,3}  3∇ ≡ 7∇^{S3,2}  1∇ ≡ 3∇^{S4,3} 
For the dozenscores:
If Z reduces to 0∇ and Y reduces to 1∇, then Z and Y taken together form a GraecoLatin 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 dozenscores can be partitioned into six subsets (quadragintas) of 40 operations each, as shown in the tables below.


 



The following formulas show more specifically how the dozenscores 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} 
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:
With linear operations, four relations can never occur, because they would be connected with a violation of FOC or SOC:
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 240^{4} possibilities) of operations that satisfy this fouroperation 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 240^{4}) 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 240^{2}) 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 S_{5} = { 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 secondorder cancellative:
For each of S_{7} and S_{11}, there are two substantively different configurations. Examples, in canonical form, appear below preceded by 1∇ for comparison:
 
1∇(a, b) = 1∇(a + 1, b + 2) 

 
U_{1}(a, b) = U_{1}(a + 1, b + 2)  U_{2}(a, b) = U_{2}(a + 1, b + 3) 

 
U_{3}(a, b) = U_{3}(a + 1, b + 2)  U_{4}(a, b) = U_{4}(a + 1, b + 3) 
All the other FOC and SOC operations over S_{7} and S_{11} can be found by making the obvious adaptations to the following transformations that were defined for S_{5}:
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 S_{13} that, although it is in canonical form, is conspicuously nonlinear:
 
U_{5}(a, b) = U_{5}(a + m, b + n) cannot be satisfied 
§ 8. The threeoperand case can also be glimpsed. The following definitions are practically inevitable, and assume modulon arithmetic when the operation is over S_{n}:
FOC is satisfied when:
SOC is satisfied when:
Thirdorder cancellativity (TOC) is satisfied when:
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 facecentered and bodycentered cubic lattices.)
Some existences: