On iterating semiproper preorders

Journal of Symbolic Logic 67 (4):1431-1468 (2002)
Let T be an $\omega_{1}-Souslin$ tree. We show the property of forcing notions; "is $\lbrace\omega_{1}\rbrace-semi-proper$ and preserves T" is preserved by a new kind of revised countable support iteration of arbitrary length. As an application we have a forcing axiom which is compatible with the existence of an $\omega_{1}-Souslin$ tree for preorders as wide as possible
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DOI 10.2178/jsl/1190150293
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