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. We investigate the relation between the theory of the itera- tions 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 cer- tain 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.

Compatibility of Shelah and Stupp's and of Muchnik's iteration with fragments of monadic second order logic