Compositions of dozenscores.
Version of Sunday 27 January 2013.
§ 6d1. In algebra and analysis, the composition of unary operations is frequently studied. However, composition may easily be applied to binary operations such as the dozenscores, yielding a structure analogous to the binary tree of computer science.
Needed are two variables a and b, along with a supply of operations V_{0}, V_{1}, and so forth; some of these V_{n} might be equal. For this purpose, infix notation is clearer than prefix, and the parentheses are significant because the dozenscores are rarely associative. In the table of examples below, each insertion is shown in red for emphasis.
building trees  

Here is an ordinary invocation of a binary operation:  a V_{0} b  #1  
Within #1, (a V_{1} b) can be placed two ways:  for the a:  (a V_{1} b) V_{0} b  #2a 
for the b:  a V_{0} (a V_{1} b)  #2b  
Within #2b, (a V_{2} b) can be placed three ways:  for the left a:  (a V_{2} b) V_{0} (a V_{1} b)  #3a 
for the right a:  a V_{0} ((a V_{2} b) V_{1} b)  #3b  
for the b:  a V_{0} (a V_{1} (a V_{2} b))  #3c  
From #3c, examples of further successive substitutions are:  a V_{0} ((a V_{3} b) V_{1} (a V_{2} b))  #4  
((a V_{4} b) V_{0} ((a V_{3} b) V_{1} (a V_{2} b))  #5  
(a V_{4} b) V_{0} ((a V_{3} b) V_{1} ((a V_{5} b) V_{2} b))  #6  
((a V_{6} b) V_{4} b) V_{0} ((a V_{3} b) V_{1} ((a V_{5} b) V_{2} b))  #7 
Note that in these constructions:
In the parlance of binary trees, the a's and b's serve in the role of leaf nodes.
§ 6d2. In most cases the following compositions, which are the simplest nontrivial trees:
(a V_{1} b) V_{0} b
a V_{0} (a V_{1} b)
do not yield dozenscores. For instance, below are two ordinary dozenscores followed by four examples of compositions thereof. In the tables of those operations that are not dozenscores, note that the colors are reversed and a delta serves as an upsidedown nabla; these differences in presentation help prevent mistakes in interpreting the tables.

 



 
(a 57∇ b) 22∇ b  a 22∇ (a 57∇ b)  (a 22∇ b) 57∇ b  a 57∇ (a 22∇ b) 
Although the delta tables above may seem random, two things can be said:
Some compositions exhibit partial or full cancellativity:



 
a 85∇ (a 173∇ b) FOC but not SOC  (a 151∇ b) 117∇ b SOC but not FOC  a 63∇ (a 66∇ b) SOC but not FOC  a 145∇ (a 167∇ b) FOC and SOC 
If a and b vary independently throughout all values of S_{5}; and if V_{0}, V_{1} and V_{R} are dozenscores being sought:
quadraginta A  

1  2  3  4  5  
1  0∇  192∇  180∇  129∇  66∇ 
2  110∇  59∇  47∇  239∇  173∇ 
3  167∇  101∇  89∇  26∇  218∇ 
4  213∇  150∇  138∇  72∇  21∇ 
5  181∇  128∇  67∇  1∇  193∇ 
6  238∇  172∇  111∇  58∇  46∇ 
7  20∇  212∇  151∇  139∇  73∇ 
8  100∇  88∇  27∇  219∇  166∇ 
In the case of V_{0} ≡ V_{1}, everything must come from quadraginta A. In each column of the table below are listed, as ordered pairs, all ten solutions for one of the two compositions. Each nabla is subscripted with the row and column numbers of where the operation is found in the quadraginta A table. This labeling makes the patterns more conspicuous.
(a V_{0} b) V_{0} b = a V_{R} b  a V_{0} (a V_{0} b) = a V_{R} b 
V_{0}, V_{R}  V_{0}, V_{R} 
0∇_{11}, 27∇_{83}  20∇_{71}, 47∇_{23} 
192∇_{12}, 100∇_{81}  212∇_{72}, 110∇_{21} 
180∇_{13}, 219∇_{84}  151∇_{73}, 239∇_{24} 
129∇_{14}, 88∇_{82}  139∇_{74}, 59∇_{22} 
66∇_{15}, 166∇_{85}  73∇_{75}, 173∇_{25} 
238∇_{61}, 138∇_{43}  213∇_{41}, 111∇_{63} 
172∇_{62}, 213∇_{41}  150∇_{42}, 238∇_{61} 
111∇_{63}, 72∇_{44}  138∇_{43}, 58∇_{64} 
58∇_{64}, 150∇_{42}  72∇_{44}, 172∇_{62} 
46∇_{65}, 21∇_{45}  21∇_{45}, 46∇_{65} 
Whenever V_{0} and V_{R} come from the same column of the quadraginta table, it is column 5. Thus the distinguished quadraginta has a distinguished column. The idempotent dozenscores 21∇ and 46∇ form the one pair that works both ways, as listed in the last row.
Given dozenscores Z and Y:
If a Z (a Y b) = a X b then (a Y^{T} b) Z^{T} b = a X^{T} b, and this applies whether Z, Y and X are dozenscores or other operations.
§ 6d3. When three dozenscores are composed, the range of possibilities grow far beyond what two can offer. This section looks at the balanced tree configuration:
(a V_{1} b) V_{0} (a V_{2} b)
In the following examples, some values occur more often than others within a table. Below each table is its formula, and its census:


 
(a 42∇ b) 24∇ (a 155∇ b) 25·2  (a 239∇ b) 39∇ (a 30∇ b) 15·1, 5·3, 5·4  (a 224∇ b) 10∇ (a 183∇ b) 10·0, 10·4, 5·2 
A table with either of these censuses:
There are three censuses that do not occur:
However, operations that have five instances of each value are plentiful and come in a variety of patterns not easy to categorize:


 
(a 58∇ b) 35∇ (a 78∇ b)  (a 49∇ b) 38∇ (a 197∇ b)  (a 40∇ b) 43∇ (a 60∇ b)  


 
(a 27∇ b) 59∇ (a 125∇ b)  (a 53∇ b) 68∇ (a 229∇ b)  (a 66∇ b) 32∇ (a 110∇ b) 
In the special case where a V_{3} b = (a V_{1} b) V_{0} (a V_{1} b), in other words where V_{1} ≡ V_{2}, it turns out that:
This composition is wellenough behaved that it might deserve to be called the standard composition for dozenscores and receive a special symbol:
a (V_{0} ◊ V_{1}) b = (a V_{1} b) V_{0} (a V_{1} b)
More generally, let T and U be tree expressions of the type in the "building trees" table above. Then what T ◊ U means is to substitute the expression "(a U b)" for each instance of a or b in T. Hence this unsurprising result:
a ((V_{0} ◊ V_{1}) ◊ V_{2}) b =
a (V_{0} ◊ V_{1} ◊ V_{2}) b =
a (V_{0} ◊ (V_{1} ◊ V_{2})) b =
((a V_{2} b) V_{1} (a V_{2} b)) V_{0} ((a V_{2} b) V_{1} (a V_{2} b))
The relation among the first three members of the equation above can more succinctly be written:
(V_{0} ◊ V_{1}) ◊ V_{2} ≡ V_{0} ◊ V_{1} ◊ V_{2} ≡ V_{0} ◊ (V_{1} ◊ V_{2})
As in the case of unary operations, this binary composition is associative. It also works for asymmetric trees. For instance, if T and U are thus:
a T b = ((a V_{2} b) V_{1} b) V_{0} (a V_{1} b)
a U b = a V_{3} (a V_{4} (a V_{5} b))
Then a (T ◊ U) b equals the following complicated result, with the Usubstitutions in red:
(((a V_{3} (a V_{4} (a V_{5} b))) V_{2} (a V_{3} (a V_{4} (a V_{5} b)))) V_{1} (a V_{3} (a V_{4} (a V_{5} b)))) V_{0} ((a V_{3} (a V_{4} (a V_{5} b))) V_{1} (a V_{3} (a V_{4} (a V_{5} b))))
It might be observed that supplying the same expression to the left and right leaf nodes of T turns it rather into a de facto unary function.
§ 6d4. Two more compositions involving three operations are in straight configurations:
((a V_{2} b) V_{1} b) V_{0} b
a V_{0} (a V_{1} (a V_{2} b))
which are equivalent if the transposes of the operations are taken, and if a and b are exchanged.
Each of the four columns in the table below is an example of building up, step by step, a composition in the straight configuration. Some of the intermediate and final results are dozenscores, and others are not.
— first step —  




 
a 216∇ b  a 88∇ b  a 173∇ b  a 59∇ b  
— second step —  



 
(a 216∇ b) 126∇ b  (a 88∇ b) 151∇ b  (a 173∇ b) 224∇ b  (a 59∇ b) 219∇ b  
— third step —  



 
((a 216∇ b) 126∇ b) 83∇ b  ((a 88∇ b) 151∇ b) 91∇ b  ((a 173∇ b) 224∇ b) 85∇ b  ((a 59∇ b) 219∇ b) 87∇ b 
When V_{0}, V_{1} and V_{2} are any dozenscores, the possible censuses of the operations at the three steps forming the straight configuration are the same as in the balanced case above, namely:
More can be said about the following straight configuration:
a V_{R} b = ((a V_{2} b) V_{1} b) V_{0} b
where V_{0}, V_{1} and V_{2} are dozenscores, as usual. If V_{R} is a dozenscore, then V_{1} must be from quadraginta A.
The following list of ten dozenscores comes from rows 4 and 8 of quadraginta A:
213∇  150∇  138∇  72∇  21∇ 
100∇  88∇  27∇  219∇  166∇ 
When V_{1} is in this list:
If V_{0} ≡ V_{1} ≡ V_{2}, then there are ten solutions, all from the list of ten selections from quadraginta A above.
§ 6d5. Two further threeoperation compositions are in angled configurations:
a V_{0} ((a V_{2} b) V_{1} b)
(a V_{1} (a V_{2} b)) V_{0} b
which themselves are equivalent if the transposes of the operations are taken, and if a and b are exchanged, as before.
However, no simple conversion between the straight and angled configurations is available. The complication is that:
A few examples are:




 
a174∇ ((a 21∇ b) 0∇ b) 25·3  a153∇ ((a 212∇ b) 48∇ b) 5·0, 15·1, 5·3  a142∇ ((a 11∇ b) 59∇ b) 10·1, 5·3, 10·4  a185∇ ((a 234∇ b) 27∇ b) 5·0, 5·1, 5·2, 5·3, 5·4  a166∇ ((a 26∇ b) 59∇ b) 5·0, 5·1, 5·2, 5·3, 5·4  




 
a101∇ ((a 142∇ b) 107∇ b) 6·0, 6·1, 1·2, 11·3, 1·4  a103∇ ((a 158∇ b) 233∇ b) 6·0, 1·1, 6·2, 6·3, 6·4  a112∇ ((a 23∇ b) 104∇ b) 2·0, 2·1, 7·2, 7·3, 7·4  a124∇ ((a 214∇ b) 102∇ b) 3·0, 3·1, 8·2, 3·3, 8·4  a135∇ ((a 23∇ b) 109∇ b) 4·0, 4·1, 9·2, 4·3, 4·4 