"Embedding" a language in itself
This question is related to the so called insertion systems.

An insertion system is a special type of rewriting system whose rules are of the form $1 \rightarrow r$ for all $r$ in a given language $R$. Let us write $u \rightarrow_R v$ if $u = u'u''$ and $v = u'ru''$ for some $r \in R$. Let us denote by $\buildrel{*}\over\rightarrow_R$ the reflexive transitive closure of the relation $\rightarrow_R$. The closure of a language $L$ of $A^*$ under $\buildrel{*}\over\rightarrow_R$ is the language
$$  
[L]_{\buildrel{*}\over\rightarrow_R} = \{ v \in A^* \mid \text{ there exists $u \in L$ such that $u \buildrel{*}\over\rightarrow_R v$} \}
$$
Recall that a well quasi-order on a set $E$ is a reflexive and
transitive relation $\leqslant$ such that for any infinite sequence $x_0,
x_1, \ldots$ of elements of $E$, there are two integers $i &lt; j$ such
that $x_i \leqslant x_j$. The following theorem is proved in [1]:


  If $H$ is a finite set of words such that the language $A^* \setminus
A^*HA^*$ is finite, then the relation $\buildrel{*}\over\rightarrow_R$ is a well quasi-order on $A^*$ and $[L]_{\buildrel{*}\over\rightarrow_R}$ is regular.


[1] W. Bucher, A. Ehrenfeucht and D. Haussler, On total regulators generated by derivation relations, Theor. Comput. Sci. 40, 2-3 (1985), 131– 148.