# Transform a non-regular language into a regular one using sort

Is there a way where sort turns a non regular language into a regular one. What I mean by sort is this: Consider the language $$L =$$ { $$bac, cbca, acbb$$}. $$sort(L) =$$ {$$abc, abcc, abbc$$} respectively. If there is, do you have an example. If not can you provide a proof if you can.

• Can you define the operator in more precise terms, do you mean writing all zeroes before ones and writing all as before bs before cs etc? Mar 11 at 12:33
• There is no order within a set. Mar 11 at 13:29
• @Anonymous please edit the question to include what sort means instead of describing it in a comment Mar 11 at 13:47
• @Russel. I reformulated my question. Mar 11 at 13:54
• Sort may also do the opposite: changing a regular language into a nonregular one: from $(abc)^*$ into $\{a^nb^nc^n\mid n\ge 0 \}$. Mar 12 at 1:41

No the class of all non-regular languages is not closed under sort. For example take the language $$L=\{a^nba^n\}$$. It is clearly not regular (by the pumping lemma). However, $$sort(L)$$ is the language defined by even many $$a$$s and then a $$b$$. This language is clearly regular.