Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected?
An issue Duffus raised in 1978
Jonathan David Farley
Received January 19, 2022. Published online August 29, 2022.
Abstract: Duffus wrote in his 1978 Ph.D. thesis, "It is not obvious that $P$ is connected and $P^P\cong Q^Q$ imply that $Q$ is connected", where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P\cong Q$.
Affiliations: Jonathan David Farley, Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, United States of America, e-mail: lattice.theory@gmail.com