An Issue Raised in 1978 by a Then-Future Editor-in-Chief of the Journal "Order": Does the Endomorphism Poset of a Finite Connected Poset Tell Us That the Poset Is Connected?

In 1978, Dwight Duffus—editor-in-chief of the journal “Order” from 2010 to 2018 and chair of the Mathematics Department at Emory University from 1991 to 2005—wrote that “it is not obvious that $P$ is connected and $P^P$ isomorphic to $Q^Q$ implies that $Q$ is connected,” where $P$ and $Q$ are finite non-empty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P\cong Q$.