Extensions of Language Families and Canonical Forms for the Corresponding AFL-structures

By restrictions on the definition of hyper-algebraic extension we obtain a few other extension operators $X$, which transform a language family $K$ into an "enriched" family $X(K)$. Well-known AFL-structures (such as full AFL, semi-AFL, super-AFL, substitution-closed AFL, etc.) are characterized by means of full $X$-AFL's, i.e. nontrivial families closed under (i) finite substitution, (ii) intersection with regular sets, and (iii) the operator $X$. For the least full $X$-AFL $\hat{\cal X}(K)$ containing $K$, we establish Canonical Forms, i.e. we decompose the operator $\hat{\cal X}$ into a single product of the simpler operators $X$ and $\Pi$, where $\Pi(K)$ is the least nontrivial family containing $K$ and closed under (i) and (ii).