Structural Similarity in Context-Free Languages

Divide the context-free languages into equivalence classes in the following way: L1 and L2 are in the same class if there are a-transducers M and M such that M(L1) ~ L~ and d~r(L,) = L1. Define L1 and L~ to be structurally similar if they are in the same class. Among the results given below are: (I) if L1 and L2 are structurally similar and L1 has a structurally similar set of (right) sententiat forms then so does L2 ; (2) i fL 1 and L2 are structurally similar and L1 is deterministic, then L2 has a structurally similar set of right sentential forms; (3) ilL1 and L2 are structurally similar and L1 is a parenthesis language then Le has a structurally similar set of sentential forms; (4) there is a nonempty equivalence class of structurally similar languages that contains no (right) sentential forms of any grammar; (5) if an equivalence class contains any set of (right) sentential forms at all then every language in the class has a set of (right) sentential forms in that class.