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A transformation of structures 驴 is monadic second-order compatible (MS-compatible) if every monadic second-order property P can be effectively rewritten into a monadic second-order property Q such that, for every structure S, if T is the transformed structure 驴(S), then P(T) holds iff Q(S) holds.We will review Monadic Second-order definable transductions (MS-transductions): they are MS-compatible transformations of a particular form, i.e., defined by monadicsec ond-order (MS) formulas.The unfolding of a directed graph into a tree is an MS-compatible transformation that is not an MS-transduction.The MS-compatibility of various transformations of semantical interest follows. We will present three main cases and discuss applications and open problems.