The left/right ALM linear content (LC) algebras for chapter 2 were constructed from the corresponding (co)bif-2 algebras with the three special verses, 2:{0, 1, 286}, serving as the basis. The LC algebras facilitate algebraic manipulations of the verses LCs in the chapter, providing new inter-relations among verses.
The LC algebra provides a general framework to perform algebraic operations on the verse contents. However, in order to restrict the focus on any particular verse to further study its algebraic properties, it is helpful to introduce linear operators whose actions on the verse-contents can uncover additional structures associated with the verse. The LC algebras were defined via their multiplication tables with no further constraints or restrictions. Subsequently, the multiplication is not required to posses any specific properties. In particular, they are neither commutative nor associative: UV ≠ VU and (UV)W ≠ U(VW) for LCs U, V, and W. For instance, in the left LC algebra, AL = (10518, 10597, 9284), while LA = (1668, 1665, 5). Similarly, (AL)M = (5988, 5485, 5143), whereas A(LM) = (3678, 2947, 9470). Additionally, there is no multiplicative neutral (identity) element I such that IU = UI = U, for all U in the algebra.
Let [U, V] := UV − VU and [U, V, W] := (UV)W − U(VW) be defined as the commutator and the associator. They respectively are the measure of the deviations of the algebra from commutativity and associativity. When [U, V] = 0 for all U and V, the algebra is commutative. Similarly, if [U, V, W] = 0 for all U, V, and W it means the algebra is associative. When the multiplication is not associative, bracketing or parenthesization becomes necessary in product expressions with more than two operands. There are two ways to parenthesize the product of three elements: (UV)W and U(VW), five ways to parenthesize the product of four elements: ((WX)Y)Z, (W(XY))Z, (WX)(YZ), W((XY)Z), W(X(YZ)). The products of five elements involve 14 different bracketings and so on. In general, the number of ways products of n elements can be bracketed is the Catalan number Cn. They have numerous geometric and combinatorial properties.
To have a better handle on the set of these bracketings it is beneficial to define linear operators to formally manipulate them. To that end, it is more appropriate to recast the product expressions in forms that eliminate the need for parentheses. Product expressions are normally written in infix notations: U * V where the product sign appears between the two operands U and V, (usually it is omitted and simply written as UV). In this form bracketing is required for longer expressions. However, the same product can be expressed in postfix form where the operands appear first followed by the product sign: U V *. For instance the infix notation (WX)(YZ) written explicitly as (W * X) * (Y * Z) becomes W X * Y Z * *. It is executed from left to right: it takes the first two operands W and X then do their product (*) then take the second pair Y and Z and similarly do their product (*) then multiply the two partial products. There is also the symmetric counterpart prefix form where the multiplication sign appears first followed by operands.
Since, the bracketing itself is of prime interest and not the operands (U, V, etc.) or even the nature of any particular multiplication (only that it is nonassociative) the expressions can be simplified where (WX)(YZ) becomes (⋅ ⋅)(⋅ ⋅). Here, the "⋅"s stand for the operands W, X, Y, and Z. The products are represented by concatenation: ⋅ ⋅. The corresponding postfix expression is ⋅ ⋅ * ⋅ ⋅ * *. The simplified postfix expression contains only two symbols, ⋅ and *. For convenience symbol 1 can be used instead of ⋅ and 0 for *. The resulting binary expression or bitstring 1101100 is the final postfix representation of (⋅ ⋅)(⋅ ⋅). The bitstring 1101100 can equivalently be written in hexadecimal format 0X6C. Recal, the 16 digits in hexadecimal base are 0 - 9, A - F. The six letters A - F stand for digits 10 - 15. The 0X at the beginning indicates it is in hexadecimal base.
A postfix bitstring (PB) is always of odd length with 1s outnumbering 0s by one. This is because the number of operands is one more than the number of multiplications. For instance in (W * X) * (Y * Z) there are four operands and three multiplications, so its PB contains four 1s and three 0s, (1101100), therefore of length 7. When traversing a PB from left to right, the 1s always appear more frequently than the 0s. In general, an arbitrary bitstring where the 1s occur one more than the 0s can be turned into a PB with a unique cyclic shift or rotation.
PBs allow defining linear operators to formally manipulate bracketings.
Given a PB, the ith complement operator Ci is
defined to act on a PB, resulting in a new PB that is obtained by holding
the ith 1 fixed while complementing all other bits followed by
a unique rotation, if necessary. Recall, in base two, the binary
complementation flips the bits where 1s become 0s and vice a versa. As an
example, the C1 action on (⋅ ⋅)(⋅ ⋅) is
obtained as follows:
C1 (⋅ ⋅)(⋅ ⋅)
= C1 1101100 ⇒ 1010011 ⇒
1110100 = ⋅((⋅ ⋅) ⋅).
The bracket
expression (⋅ ⋅)(⋅ ⋅) is first replaced by its
equivalent PB, 1101100. Then its first 1 is held fixed while the remaining
bits are flipped (complemented) resulting in the second bitstring which is
not yet a PB (the 1s must outnumber the 0s from left to right). After the
necessary rotation it turns into the PB 1110100 with its equivalent
infix form ⋅((⋅ ⋅) ⋅).
The linear bracket operator Tij acts on a PB by transposing
(swapping) the ith 1 with jth 0 followed by any
necessary rotation. Here is the action of T42 on the same
PB:
T42 (⋅ ⋅)(⋅ ⋅) =
T42 1101100 ⇒ 1101010 = ((⋅ ⋅)
⋅) ⋅.
Here, the 4th 1 swaps place with
2nd 0 which happens to be a PB, (no rotation needed), with its
infix form ⋅((⋅ ⋅) ⋅).
The operator Dij deletes the ith 1 and
jth 0 resulting in a PB of length shortened by two. The action
of D22 on (⋅ ⋅)(⋅ ⋅):
D22
(⋅ ⋅)(⋅ ⋅) = D22 1101100
⇒ 10110 ⇒ 11010 = (⋅
⋅) ⋅.
The final expression is of length 5.
The last operator Sij duplicates the ith 1 and
jth 0 resulting in a PB lengthen by two. The action of
S31:
S31 (⋅ ⋅)(⋅ ⋅)
= S31 1101100 ⇒ 110011100 ⇒
111001100 = (⋅ (⋅ ⋅))(⋅ ⋅),
resulting in an expression of length 9.
The commutators/associators along with the bracket operators defined above provide the mechanism to direct the focus exclusively to a particular verse in order to study the algebraic structures of its linear content or appropriate subsets of it.