Full AFLs and nested iterated substitution

A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set. Every superAFL is a full AFL containing all context-free languages. If L is a full principal AFL, then Ŝ∞(L), the least superAFL containing L, is full principal. If L is not substitution closed, the substitution closure of L is properly contained in Ŝ∞(L). The indexed languages form a superAFL which is not the least superAFL containing the one-way stack languages. If L has a decidable emptiness problem, so does Ŝ∞(L). IfDs is an AFA, L = L(Ds) and Dw is the family of machines whose data structure is a pushdown store of tapes of Ds, then L(Ds) = Ŝ∞(L) if, and only if, Ds is nontrivial. If Ds is uniformly erasable and L(Ds) has a decidable emptiness problem, then it is decidable if a member of Dw is finitely nested