In other words, each string in is the prefix of a string in , with the remainder of the word being a string in .
Now, if we insert a divider into the middle of an element of , the part on the right is in only if the divider is placed adjacent to a b (in which case i ≤ n and j = n) or adjacent to a c (in which case i = 0 and j ≤ n). The part on the left, therefore, will be either or ; and can be written as
Some common closure properties of the right quotient include:
- The quotient of a regular language with any other language is regular.
- The quotient of a context free language with a regular language is context free.
- The quotient of two context free languages can be any recursively enumerable language.
- The quotient of two recursively enumerable languages is recursively enumerable.
Left and right quotients
There is a related notion of left quotient, which keeps the postfixes of without the prefixes in . Sometimes, though, "right quotient" is written simply as "quotient". The above closure properties hold for both left and right quotients.