Is the set of "endomorphisms" of a directed set again a directed set?

Question

Let $(I,\leq)$ be a directed set, that is $\leq$ is reflexive and transitive and for every $a,b\in I$ we find $c\in I$ such that $a,b\leq c$. Now consider the set $M$ consisting of all maps $\sigma:I\longrightarrow I$ such that $a \leq b$ implies $\sigma a≤ \sigma b$. We define a reflexive and transitive order on $M$ as follows: For $\sigma,\tau:I⟶I$ in M we set $\sigma \leq \tau$ if for all $a\in I$ we obtain $\sigma a \leq \tau a$. Now the question is: Is $M$ again a directed set?

Answer 1

I claim that the answer is no. This answer is a generalization of the answers to this previous question.

If $X,Y$ are posets, then let $\text{Hom}(X,Y)$ denote the set of all mappings $f:X\rightarrow Y$ where if $x_{1},x_{2}\in X$ and $x_{1}\leq x_{2}$, then $f(x_{1})\leq f(x_{2})$.

Theorem: The following are equivalent:

1. $\text{Hom}(X\times X,X\times X)$ is directed.

2. $\text{Hom}(X\times X,X)$ is directed.

3. $\text{Hom}(P,X)$ is directed for each poset $P$.

4. There is a function $L:X\times X\rightarrow X$ such that $x\leq L(x,y),y\leq L(x,y)$ for each $x,y\in X$ and where if $x_{1}\leq x_{2},y_{1}\leq y_{2}$, then $L(x_{1},y_{1})\leq L(x_{2},y_{2})$.

Proof:

$3\rightarrow 2$ is trivial.

$4\rightarrow 3$ Suppose that $f,g\in\text{Hom}(P,X)$. Then let $h:P\rightarrow X$ be the function defined by $h(p)=L(f(p),g(p))$. Then $h\in\text{Hom}(P,X)$ and $f\leq h,g\leq h$.

$1\leftrightarrow 2$ Observe that $\text{Hom}(X\times X,X\times X)\simeq \text{Hom}(X\times X,X)\times\text{Hom}(X\times X,X)$. To get the equivalence observe that two posets $P,Q$ are both directed if and only if the product $P\times Q$ is direted.

$2\rightarrow 4$ Let $\pi_{0},\pi_{1}:X\times X\rightarrow X$ be the projections where $\pi_{0}(x,y)=x,\pi_{1}(x,y)=y$. Then there is some $L:X\times X\rightarrow X$ where $\pi_{0}\leq L,\pi_{1}\leq L$. Q.E.D.

In the previous question, we have an example of posets $P,Q$ where $Q$ is directed by where $\text{Hom}(P,Q)$ is not directed. In this case, $\text{Hom}(Q\times Q,Q\times Q)$ is not directed.

As Joel David Hamkins answered in the previous question, if $X$ is a countable directed set, then $\text{Hom}(P,X)$ is always directed. More generally, if $X$ has a linearly ordered cofinal subset or if $X$ is a join semi-lattice, then $\text{Hom}(P,X)$ is directed.

Let me now generalize and slightly modify Emil Jeřábek's counterexample to the previous question and obtain a directed poset $Y$ where $\text{Hom}(P,Y)$ is not directed for some poset $P$.

Let $\lambda,\kappa$ be regular cardinals with $\lambda<\kappa$. If $\Delta$ is a set, then let $P_{<\lambda}(\Delta)$ be the collection of all subsets of $\Delta$ of cardinality less than $\lambda$, and order $P_{<\lambda}(\Delta)$ by inclusion. Let $G$ be a two element poset with underlying set $\{r,s\}$ where $r\not\leq s,s\not\leq r$. Let $X=\kappa\times G$ be the cartesian product. Now let $Y=X\cup P_{<\lambda}(\kappa)$ be the partial ordering where if $(\alpha,t)\in X,A\in P_{<\lambda}(\kappa)$, then $(\alpha,t)\leq A$ if and only if $\alpha\leq\alpha'$ for some $\alpha'\in A$. Then the poset $Y$ is directed.

Then let $f,g:\kappa\rightarrow Y$ be the mappings where $f(\alpha)=(\alpha,r),g(\alpha)=(\alpha,s)$. Then there cannot be a $h\in\text{Hom}(\kappa,Y)$ with $f\leq h,g\leq h$. Suppose there were such an $h$. Then we would have $h(\alpha)\in P_{<\lambda}(\kappa)$ for each $\alpha\in\kappa$. Furthermore, since $(\alpha,r)=f(\alpha)\leq h(\alpha)$, we know that $\alpha\leq\alpha'$ for some $\alpha'\in h(\alpha)$. Therefore, the sequence $h$ cannot be eventually constant. Thus, $h$ is a not-eventually constant monotone sequence in $P_{<\lambda}(\kappa)$ of cofinality $\kappa$ which is impossible.