How to check for a string that for every character in it, there exists all the characters which are alphabetically smaller than it before it  e.g aab  is correct while aacb is not, because the second case, we have 'c' but 'b' is not present before it. 
Also aac is not correct as it does not have 'b' before 'c'.
I believe to understand your question correctly, and here is my attempt at answering it, if I have mis-understood please correct me.
The standard comparisons (<, <=, >, >=, ==, !=) apply to strings. These comparisons use the standard character-by-character comparison rules for ASCII or Unicode.  That being said, the greater and less than operators will compare strings using alphabetical order.
a string that for every character in it, there exists all the characters which are alphabetically smaller than it before it