Conjugacy relations of prefix codes

It is shown that, if X and Y are prefix codes and Z   is a non-empty language satisfying the condition XZ=ZYXZ=ZY, then Z   is the union of a non-empty family {Pn}i∈I{Pn}i∈I of pairwise disjoint prefix sets such that XPi=PiYXPi=PiY for all i∈Ii∈I. Consequently, the conjugacy relations of prefix codes are explored and, under the restriction that both of X and Y   are prefix codes, the solutions of the conjugacy equation XZ=ZYXZ=ZY for languages are determined. Also, the decidability of the conjugacy problem for finite prefix codes is confirmed.