How are "bitloops", "links" and "chains" more formally described? I've done some independent work on partitions of powersets and want to know how it would be described more formally, or in what directions I might study to help further develop it in the correct language. More precisely, can "bitloops", "links" and "chains" be described without making up my own words for them? Here is the context:

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*Let a powerset $P(S)$ be described by bitstrings, so the bits determine what elements of S are included in a given element of $P(S)$.


*Partition $P(S)$ into rotation classes, for example, {100, 010, 001} is a rotation class and each bitstring is a rotation of the other.


*Define a bitloop as a bitstring whose ends are connected to form a loop, so it can represent a rotation class.


*Define a unary operation, the link. This takes a bitloop and produces another by performing addition modulo 2 on adjacent bits. For example the link of the bitloop 101010 (where the bits on the ends are also adjacent) is 111111, because 1 + 0 = 1. Call a "sublink" the reverse operation (which may produce 0, 1 or 2 bitloops).


*Define a chain as a set of bitloops equipped with the link operation, such that for any bitloop in a chain, the link of that bitloop is also included, and so are its sublinks, if they exist.


*Partition the bitloops of $P(S)$ into chains.


*Chains can be represented by graphs, where bitloops are the vertices and links are the edges. Chains can be isomorphic, meaning they share the same graph. So $P(S)$ can be further partitioned by graphs.
 A: Here is a theoretical framework as you desire it.
If $n$ is the size of these strings, the "ambient" vector space $V:=\mathbb{Z}_2^n=\{0,1\}^n$ which is isomorphic to power set $P(S)$ (with, in particular, the same number of elements $2^n$)
In this vector space you can operate with matrices.
For example, the shift operator $S:x\to x+1$ has the following matrix :
$$S=\left(\begin{array}{llllll}
0&1&0&0&0&0\\
0&0&1&0&0&0\\
0&0&0&1&0&0\\
&&&\cdots&&\\
0&0&0&0&0&1\\
1&0&0&0&0&0\\
\end{array}\right)$$
A rotation class is the set $\{s,Ss,S^2s,... S^{n-1}s\}$
The link operator is clearly given by this matrix :
$$L=\left(\begin{array}{llllll}
1&1&0&0&0&0\\
0&1&1&0&0&0\\
0&0&1&1&0&0\\
&&&\cdots&&\\
0&0&0&0&1&1\\
1&0&0&0&0&1\\
\end{array}\right)=I+S$$
therefore is a linear operator (which wasn't evident "a priori").
(where $I$ is the $n \times n$ identity matrix).
What you call the sublink operator should be defined as the inverse of matrix $L$. But matrix $L$ is not invertible ... Therefore, speaking of sublink as "reverse operation" is confusing. As the rank of matrix $L$ is $n-1$, there is some hope by looking for pseudo-inverses.
One of them is
$$L_+=\left(\begin{array}{llllll}
0&1&1&1&1&1\\
1&0&1&1&1&1\\
1&1&0&1&1&1\\
&&&\cdots&&\\
1&1&1&1&0&1\\
1&1&1&1&1&0\\
\end{array}\right)=\textbf{1}\textbf{1}^T-I$$
(where $\textbf{1}$ is the column vector $n \times 1$ with all its entries equal to $1$).
with
$$L \times L_+=L \times (\textbf{1}\textbf{1}^T-I)=\underbrace{(L \times \textbf{1})}_{0}\textbf{1}^T-L= L=I\color{red}{+S} \ \ \text{instead of} \ \ L \times L_+=I.$$
(Please note that $L_+$ has full rank).
Important: In fact, the added term $\color{red}{S}$ above isn't harmful and can be suppressed if one considers that the result is "up to shifts" $S$, i.e., if one works in a given rotation class.
