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We show that for two quivers without oriented cycles related by a BGPreflection, the posets of their cluster tilting objects are related by a simplecombinatorial construction, which we call a flip-flop.  We deduce that the posets of cluster tilting objects of derived equivalentpath algebras of quivers without oriented cycles are universally derivedequivalent. In particular, all Cambrian lattices corresponding to the variousorientations of the same Dynkin diagram are universally derived equivalent.

Universal derived equivalences of posets of cluster tilting objects