A class of preorders iterated under a type of RCS

Jensen formulates classes of notions of forcing in [1]. We take subproper among those. This class is wider than proper, may differ from semiproper of Shelah and is iterable under the revised countable support a la Donder according to [1]. We formulate a class of notion of forcing whose definition involves less parameters than Jensen’s subproper and show that it iterates under a type of revised countable support of [2]. We say a preorder P is preproper, if there exist a set z and a regular cardinal θ with z, P ∈ Hθ such that for all (χ,A,N, p, s) such that • χ is a regular cardinal with Hθ ⊆ Lχ[A], • (N,∈, A∩N) is a countable elementary substructure of (Lχ[A],∈, A) with z, P ∈ N , • there exists a transitive set model M of ZFC− (i.e. no power set axiom) such that the transitive collapse of N appears as a (Hτ ) with a regular τ in M ,