Does the Endomorphism Ordered Set of a Finite Ordered Set Determine the Ordered Set? The Cases of Height $1$ and "Trebled" Ordered Sets

Source abstract

For ordered sets $X$ and $Y$, let $Y^X$ denote the ordered set of order-preserving maps from $X$ to $Y$, where $f\le g$ in $Y^X$ if $f(x)\le g(x)$ for all $x\in X$. Let $P$ and $Q$ be finite ordered sets such that $P^P\cong Q^Q$. It is proven that $P\cong Q$ if $P$ or $Q$ has height at most $1$ or if $P$ and $Q$ are ordered sets of the following form: replace each element of an ordered set with a three-element antichain. The latter is an elaboration of a proof of Tim Campion.