For any string $S$ and substring $u \leqslant S$ define $n_S(u) = \max \{ n : \alpha_0 u \alpha_1 \cdots u \alpha_n = S $ for some $\alpha_i \leqslant S\}$. That is $n_S(u) = $ the maximum number of disjoint copies of $u$ that we can squeeze into $S$.
Now let $C_S = \{ u \leqslant S : |u| \geq 2 $ and $n_S(u) \geq 2 \}$.
I want to define special subsets of $C_S$, $A$, such that simultaneously $n_S(u) \geq 2$ for all $u \in A$. How can we accomplish that formula-wise without getting too complicated?
By simultaneous I mean for example $\alpha u \beta v \alpha' u \beta' v \leqslant S \implies$ "$n_S(u), n_S(v) \geq 2$ simultaneously".
Such a set of substerings $A$ is identified by:
There exists a decomposition $S = \sigma_1 \cdots \sigma_n$ such that
$$ \forall u \in A, \exists 1\leq i \lt j \leq n: \sigma_i = u = \sigma_j $$