  We investigate the relation between the theory of the iterations in the sense
of Shelah-Stupp and of Muchnik, resp., and the theory of the base structure for
several logics. These logics are obtained from the restriction of set
quantification in monadic second order logic to certain subsets like, e.g.,
finite sets, chains, and finite unions of chains. We show that these theories
of the Shelah-Stupp iteration can be reduced to corresponding theories of the
base structure. This fails for Muchnik's iteration.
