Dear All,

there is one type of regular languages, over $\{a,b\}$, which appear naturally in what I am studying, so if anybody could recognise them, or say any sort of their characterisation, that would be great.

So, our languages $L\subseteq (a+b)^*$ are defined as follows:

(0) First we choose some $n\geq 0$ and new letters $x_1,\ldots,x_n$ (if $n=0$, then we just basically don't choose any new letters);

(1) We choose words $w_i\in (a+b)^*$ for all $1\leq i\leq n$;

(2) Let $\phi$ be the homomorphism lifting the assignment $a\mapsto a$, $b\mapsto b$, $x_i\mapsto w_i$;

(3) Find a finite set $F$ from $(a+b+x_1+\cdots+x_n)^*$;

(4) Let $K=(a+b+x_1+\cdots+x_n)^{*}-(a+b+x_1+\cdots+x_n)^{*}F(a+b+x_1+\cdots+x_n)^{*}$

(5) Finally $L=\phi(K)$.

It is quite easy to see that $ba^+b$ cannot be obtained this way.