On two letters versus three

If A is a context-free language over a two-letter alphabet, then the set of all words obtained by sorting words in A and the set of all permutations of words in A are context-free. This is false over alphabets of three or more letters. Thus these problems illustrate a difference in behavior between twoand three-letter alphabets. The following problem appeared on a recent exam at Cornell: Let be a finite alphabet with a fixed total ordering on the letters. For a string x , let sort x be the string obtained by sorting the letters in increasing order. For example, if a b c, then sort abacbaa aaaabbc. For A , let sortA fsortx j x Ag. Of the following three statements, two are false and one is true. Give counterexamples for the two false ones and a proof of the true one. (i) If A is regular, then so is sort A. (ii) If A is context-free, then so is sortA. (iii) If A is context-sensitive, then so is sort A. One might also ask the same questions about perm A, the set of all permutations of words in A. Of course, it is (i) and (ii) that are false, since sort abc perm abc a b c fabc j n g