Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise.


Is there a poset with more than 1 point such that $P\cong \text{End}(P)$?



Posets isomorphic to their endomorphism poset
If $P$ is infinite and $P\cong \textrm{End}(P)$, then the size of
$P$ is restricted by the following facts.


Lemma. Assume that $P$ is an infinite poset.


1. $P$ has an infinite antichain, an infinite well-ordered subset,
or an infinite inversely well-ordered subset.
2. If $|P|>2^{\lambda}$ for some infinite $\lambda$, then $P$ has an antichain, a well-ordered subset,
or an inversely well-ordered subset of size $>\lambda$.
3. If $P$ has an antichain of size at least $\kappa\geq\omega$, then $\textrm{End}(P)$
has an antichain of size at least $2^{\kappa}$.
4. If $P$ has a well-ordered or inversely well-ordered subset
of size at least $\kappa\geq\omega$, then $\textrm{End}(P)$
has an antichain of size at least $2^{\kappa}$.


In particular, these facts imply that if $P\cong \textrm{End}(P)$,
then $|P|\geq \beth\_{\omega}$. If also 
$|P|>2^{\lambda}$ for some infinite $\lambda$, then $|P|>2^{2^{\lambda}}$.


Sketch of proofs.


Part 1 of the Lemma follows from Ramsey's Theorem.


Part 2 of the Lemma follows from the Erdos-Rado Theorem.


For part 3 of the Lemma, if $P$ is an antichain, then $\textrm{End}(P)$
is an antichain of size $2^{|P|}$, so the claim is clear. Otherwise
choose $a<b$ in $P$. Let $A\subseteq P$ be an antichain in $P$
of cardinality $\kappa$. It is possible to find a set
$\mathcal X$ of pairwise incomparable subsets of $A$ such that
$|\mathcal X|=2^{\kappa}$. For each $U\in \mathcal X$
define $f\_U\colon P\to P$ by $f\_U(x)=b$ if $x$ is in the
filter generated by $U$ in $P$, else $f\_U(x)=a$.
The set $\{f\_U | U\in \mathcal X\}$ is an antichain of size
$2^{\kappa}$ in $\textrm{End}(P)$. 


For part 4 of the lemma, assume that 
$W = \{w\_{\alpha} : \alpha < \kappa\}$ is an enumeration of
an increasing $\kappa$-sequence in $P$.
Define a partial function $\sigma\colon P\to P$ by taking
$\sigma(x)$ to be the least element $w\_{\alpha}$ in $W$
such that $x\not\geq w\_{\alpha}$. This $\sigma$ is a monotone total
function if $W$ is cofinal in $P$, but only a partial function
if there is some $q\in P$ above every $w\_{\alpha}$. If such $q$
exists, fix one such, and extend the definition of $W$ to include $w\_{\kappa}=q$
and extend the definition of $\sigma$ so that $\sigma(x)=q$
if $x\geq w\_{\alpha}$ for all $\alpha<\kappa$. Either way, $W$
is well-ordered, of size $\kappa$, and $\sigma\in \textrm{End}(P)$
maps $P$ into $W$ and $\sigma(P)$ is well-ordered and of size $\kappa$.
(Note: $\sigma(P)$ might be a proper subset of $W$.)


For every subset $V\subseteq \sigma(P)$ not containing $w\_{\kappa}$
define $g\colon \sigma(P)\to \sigma(P)$ by $g(w\_{\alpha})=w\_{\alpha}$
if $w\_{\alpha}\in \sigma(P)\setminus V$ and
$g(w\_{\alpha})=w\_{\alpha+1}$
if $w\_{\alpha}\in V$. If $V\_1$ and $V\_2$ are incomparable subsets
of $\sigma(P)$, then $g\_{V\_1}$ and $g\_{V\_2}$ are incomparable elements
of $\textrm{End}(P)$. There is a set
of $2^{\kappa}$-many pairwise incomparable
subsets of the $\kappa$-element set $\sigma(P)\setminus\{w\_{\kappa}\}$,
so $\textrm{End}(P)$ has an antichain
of size at least $2^{\kappa}$. \\


