representing set of non-overlaping string in formal notation
Idea: represent substrings by their resp. start and end index.

Let $w = w_1 \dots w_n \in \Sigma^*$ a string for some alphabet $\Sigma$. Denote (indexed) substrings of $w$ by pairs $(i,j)$ with $1 \leq i \leq j \leq n$. The actual sequence of symbols represented by $(i,j)$ would be $w_{i,j} = w_i \dots w_j \in \Sigma^*$.

Denote the set of all indexed substrings of $w$ by 

$\qquad\displaystyle S_w = \{ (wi, j) \mid 1 \leq i \leq j \leq n \} \subseteq [1..n]^2$.

Note how $\{ w_{i,j}  \mid  (i,j) \in S_w\}$ is can have fewer elements due to duplicate substrings.

Non-overlapping substrings

Two indexed substrings $(i,j), (k,l) \in S_w$ with $i \leq k$ overlap if and only if $k \leq j$. Otherwise, we call them non-overlapping.

Sets of non-overlapping substrings

A set $A \subseteq S_w$ of substrings of $w$ is non-overlapping if and only if

$\qquad \forall\, u, c \in A.\ u$ and $v$ do not overlap¹.

The set of all non-overlapping sets of substrings

The set of all non-overlapping sets of substrings is

$\qquad \mathcal{N}_w = \{ A \subseteq S_w \mid A \text{ non-overlapping²} \}$.




If you so desire, input the formal definition of "do not overlap" here, i.e. $u = w_{i,j}$ and $v = w_{k,l}$ with $i \leq k$ and $k &gt; j$. Or invent notation for the predicate "do overlap", e.g. $\operatorname{overlap}(u,v)$ or $\eta(u,v)$ or ...
Again, you can replace "non-overlapping" with the definition. I don't think that makes things more clear. The whole reason for defining shorthand is to express more complicated things easily and succinctly, after all.