The order of a regular nonempty language L is defined to be the smallest integer k for which L^k = L^k+1 if there is such a k, and otherwise.

Show that the order of L is finite if and only if there is an integer k so that L^k = L, and in this case the order of L is the smallest k such that L^k = L.

Find the order of the regular language {} {aa}{aaa}, and justify it.