Abstract: In this talk we consider all possible ways of generating, for example, the middle thirds Cantor set using an iterated function system. We will discuss some results due to Feng and Wang on the relationship between the geometry of a limit set F given by an IFS and the algebraic structure of the semi-group formed by all IFSs whose limit set is F. We will also see that self-similar subsets of Cantor sets do not have to be derived from the generating IFS.