Infinite decreasing chains in the Mitchell order

Abstract It is known that the behavior of Mitchell order substantially changes at level rank-to-rank extenders, as it ceases to be well-founded. While possible partial structure below extenders considered well understood, little about in ill-founded case. The purpose paper make a first step understanding this case, by studying extent which can ill-founded. Our main results are (i) presence extender there transitive decreasing sequence any countable length, and (ii) no such length $$\omega _1$$ ω 1 .