# Name for a special family of languages?

I was wondering whether there is a standard name in the literature for the following family $$\mathcal{F}$$ of languages over any finite alphabet $$\Sigma = \{a_1,\ldots,a_k\}$$:

$$\mathcal{F}$$ consists of all languages of the shape $$\{a_{i_1}^{c_1}a_{i_2}^{n_2}a_{i_3}^{c_3}\ldots a_{i_{k-1}}^{n_{k-1}}a_{i_k}^{c_k}: n_2,n_4,\ldots,n_{k-1} \geq 0 \wedge R(n_2,n_4,\ldots,n_{k-1})\}$$, where $$c_1,c_3,\ldots,c_k$$ are fixed nonnegative constants and $$R$$ is any fixed relation on $$m$$ variables $$u_1,u_2,\ldots,u_m$$ such that for some $$S \subseteq \{1,2,\ldots,m\} \times \{1,2,\ldots,m\}$$, $$R(u_1,u_2,\ldots,u_m)$$ holds iff $$\left((\forall i,j \in \{1,\ldots,m\})[u_i = u_j \Leftrightarrow (i,j) \in S]\right)$$.

So $$\mathcal{F}$$ contains languages like $$\{a_1^n a_2^n a_3^n: n \geq 0\}$$ (the "standard" context-sensitive but non-context-free language) and $$\{a_1^2 a_1^n a_2^m a_1 a_1^m a_2^n a_1: n,m \geq 0\}$$.

This family of languages cropped up in my work recently and I was wondering whether anything interesting could be said about it..

Thank you!

• This is a very special case of the family mentioned in cstheory.stackexchange.com/q/17976/5038. Perhaps that would be helpful for finding related literature.
– D.W.
Feb 13 '19 at 7:33