Let S be the set of sequences of the form $[a_1, a_2, ..., a_n]$, where the $a_i$'s are complex numbers and $n\in \mathbb N$. For $x,y \in S$, we say that $x$ is a repetition of $y$ if $x$ is formed by putting a number of $y$'s side by side: for example, $[1,2,3,1,2,3]$ is a repetition of $[1,2,3]$, and $[1,2,1,2,1,2,1,2]$ is a repetition of both $[1,2,1,2]$ and $[1,2]$.
Given any $x \in S$, let Sub(x) $= \{ \text{ }y \in S \text{ } |\text{ }x \text{ extends } y \}$.
Define the following relation ~:
Given $x,y \in S$, $x\text{~}y$ if the sequence of least length in Sub(x) is a cyclic permutation of the sequence of least lenght in Sub(y).
For example, $[1,2,3,1,2,3] \text{~} [2,3,1,2,3,1,2,3,1]$.
It can easily be seen that ~ is an equivalence relation on S.
My question is: Could I make this definition less wordy? Also, how could I make "x is a repetition of y" more rigorous? And is there a well-established name for S?
Actually, the set of possible values doesn't seem to play any role here, let it be $C$.
So, $S=\bigcup_nC^n$, and let $Z:=C^{\Bbb Z}=\{(c_k)_{k\in\Bbb Z}\}$ be the set of 2-way infinite sequences.
Now consider the map $f:S\to Z$ which periodically repeats the input sequence in both directions, i.e., $$f((a_0,\dots, a_{n-1})):=\ k\mapsto a_{k\mathop{mod}n}$$ And take the equivalence relation 'shifted version of', $R$ on $Z$, such that $\ (a_k)\,R\, (b_k)$ iff $\exists u\in\Bbb Z:\ \forall k\, (b_k=a_{k+u})$.
Then, for $x, y\in S$ we have $x\sim y\ \iff\ f(x)\, R\, f(y)$.
I'm not aware of any particular name for it, but we can calmly call the quotient set $S/\sim$ the set of (finite) cycles of elements of $C$.