Order homomorphism functions on $\omega_1$ I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and try again.
Let $\omega_1$ be the first uncountable ordinal,
same as the set of all countable ordinals.
$\omega_1=\{\alpha:0\le\alpha<\omega_1\} = \{\alpha:\alpha {\mathrm{\ is\ a\ countable\ ordinal}}\}$.
Let $\mathcal F$ be the set of all functions
$f:\omega_1\to\omega_1$ that
are:
(a) regressive i.e. $f(\alpha) < \alpha$ for all $0 < \alpha < \omega_1$,
and
(b) non-decreasing (same as $\le$-order-preserving),
i.e.,
if $0\le\alpha \leq \beta<\omega_1$ then $f(\alpha)\leq f(\beta)$ .
Define a partial order $\sqsubseteq$ on $\mathcal F$ by $f \sqsubseteq g$ if
$f(\alpha) \leq g(\alpha)$ for all $\alpha < \omega_1$.
Let $\mathcal K$ be the subset of $\mathcal F$, consisting of functions with
a finite range.
Formally $\mathcal K=\{f\in\mathcal F: |\{f(\alpha):\alpha<\omega_1\}|<\aleph_0\}$.
Question:
Is there a $\sqsubseteq$-order-preserving map
(same as a $\sqsubseteq$-non-decreasing map)
$\psi : \mathcal F \to \mathcal K$, i.e if $f \sqsubseteq g$
then $\psi(f) \sqsubseteq \psi(g)$, and with
the additional property that $\psi(f) \sqsupseteq f$ for all $f\in \mathcal F$ ?
Let me summarize some comments made at MO, clarifying certain partial answers.
Partial answer (A). Since every $f\in\mathcal F$ is regressive and non-decreasing, it must be eventually constant and reach its maximal value $\mu_f=\max\{f(\alpha):\alpha < \omega_1\}$. One is tempted to define $\psi(f)(\alpha)=\mu_f$ for all $\alpha$. The problem is that this is not regressive: We have $\psi(f)(\alpha)<\alpha$ only when $\alpha>\mu_f$, but I insist that $\psi(f)(\alpha)<\alpha$ whenever $0<\alpha<\omega_1$.
Partial answer (B). If we drop the requirement that $\psi$ be a $\sqsubseteq$-non-decreasing map then the answer by @NoahS below works, as well as one of my comments below, which  I move here. As above let $\mu_f=\max\{f(\alpha):\alpha < \omega_1\}$ and let $\gamma_f=\min\{\alpha:f(\alpha)=\mu_f\}$. (Then $f(\alpha)=\mu_f$ for $\alpha\ge\gamma_f$, and $f(\alpha)<\mu_f$ for $\alpha<\gamma_f$. Usually $\mu_f<\gamma_f$ unless $\mu_f=0=\gamma_f$.)
Let $\alpha_{0,f}=\mu_f$. If $\mu_f\ge1$ then let $\alpha_{1,f}=f(\alpha_{0,f})<\alpha_{0,f}$. There is a non-negative integer $n_f$ such that $\alpha_{k+1,f}=f(\alpha_{k,f})<\alpha_{k,f}$ for $k<n_f$, and $\alpha_{n_f,f}=0$. Define $\psi(f)$ as follows. If $\alpha>\alpha_{0,f}$ then let $\psi(f)(\alpha)=\alpha_{0,f}=\mu_f$. If $\alpha_{k+1,f}<\alpha\le\alpha_{k,f}$ then let $\psi(f)(\alpha)=\alpha_{k+1,f}$. (Formally also $\psi(f)(0)=0$, but in general each function in $\mathcal F$ being regressive must take value $0$ at $1$, and being non-decreasing must take value $0$ at $0$ as well.) Then $\psi(f)\in\mathcal K$ and $\psi(f)\sqsupseteq f$.
So partial answer (A) above achieves that $\psi(f)$ has a finite range, and
$\psi(f) \sqsubseteq \psi(g)$ whenever $f \sqsubseteq g$, and also $\psi(f) \sqsupseteq f$. It almost achieves that $\psi(f)$ is regressive, but not quite, and it follows that $\psi(f)$ is not in $\mathcal K$ unless $\mu_f=0$. (One could perhaps say that $\psi(f)$ is "regressive on a tail" only, which might in a different context be good enough, but the requirement in my question is that $\psi(f)(\alpha)<\alpha$ whenever $0<\alpha<\omega_1$.) On the other hand, partial answer $B$ achieves that
$\psi(f)\in\mathcal K$ (in particular both that $\psi(f)$ is regressive and has a finite range), and $\psi(f) \sqsupseteq f$ for all $f\in \mathcal F$, but not necessarily that $\psi(f) \sqsubseteq \psi(g)$ whenever $f\sqsubseteq g$. It is not clear to me if we could achieve all conditions simultaneously. Edit. Following a comment, let me clarify why in partial answer $B$ we need not have $\psi(f) \sqsubseteq \psi(g)$ whenever $f\sqsubseteq g$. Fix any ordinals $0<\beta<\delta<\nu<\omega_1$. Let $f(\alpha)=g(\alpha)=0$ if $0\le\alpha<\nu$. Let $f(\alpha)=\beta$ and $g(\alpha)=\delta$ if $\alpha\ge\nu$.
Clearly $f\sqsubseteq g$.
Then $\psi(f)(\alpha)=\beta$ if $\alpha>\beta$, and
$\psi(f)(\alpha)=0$ if $0\le\alpha\le\beta$
(where $\psi$ is as in partial answer $B$).
While $\psi(g)(\alpha)=\delta$ if $\alpha>\delta$, and
$\psi(g)(\alpha)=0$ if $0\le\alpha\le\delta$. In particular, if $\beta<\alpha\le\delta$ then $\psi(g)(\alpha)=0<\beta=\psi(f)(\alpha)$,
so $\psi(f)\not\sqsubseteq \psi(g)$.
If I were to make a guess, I would say the answer is no.
This question is an order-theoretic restatement of a question from general topology that a co-author and I considered: Whether $\omega_1$ has a monotone interior-preserving open operator $r$, that is, if $\mathcal U$ is any open cover of $\omega_1$, with the order topology, then $r(\mathcal U)$ is an interior-preserving open refinement that covers $\omega_1$, and if $\mathcal U$ refines $\mathcal V$ then $r(\mathcal U)$ refines $r(\mathcal V)$. As usual we would write $\mathcal U\preceq \mathcal V$ if $\mathcal U$ refines $\mathcal V$. In this context $f$ is intended to encode an open cover $\mathcal U(f)=\{0\}\cup\{(f(\alpha),\alpha]:\alpha<\omega_1\}$. Note that if $f\sqsubseteq g$ then $\mathcal U(g)\preceq \mathcal U(f)$.
Update Oct 19, 2018 (and May 21, 2019):
This question has now been published in a journal.
It is Question 3.2 in the following paper:
Serdica Math. J. 44 (2018) (dedicated to the memory
of Professor Stoyan Nedev (1942−2015))
ON MONOTONE ORTHOCOMPACTNESS
S.G. Popvassilev, J.E. Porter
Here is a temporary link from the editors:
http://www.math.bas.bg/serdica/2018/2018-177-186.pdf
(Update as of August 21, 2020.)
This question has been answered in the negative by Gary Gruenhage. I will post a complete answer some time in the future. Here is a sketch of the proof. The existence of an order-preserving map $\psi$ as in the question is equivalent to $\omega_1$ being monotonically orthocompact via open refinements, abbrevaited MO$_o$ (this is Theorem 3.1 in the paper, a link to which is enclosed at the end of this question). What Gary proved is that MO$_o$ implies a certain property called (A$_o$) (defined in terms of certain neignborhoods), and that $\omega_1$ does not have this property (A$_o$).
(Update April 25, 2021.)
I am about to publish an answer here with details of Gary Gruenhage's proof (thus answering the above question is the negative).
Thank you!
Original version of this post:
Let $\omega_1$ be the first uncountable ordinal,
same as the set of all countable ordinals.
Let $F$ be the set of all functions
$f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that
are (a) regressive i.e. $f(\alpha) < \alpha$ for $0 < \alpha < \omega_1$,
and (b) order-preserving (same as non-decreasing)
i.e. $f(\alpha)\leq f(\beta)$ if $\alpha \leq \beta$.
Define a partial order on $F$ by $f \leq g$ if
$f(\alpha) \leq g(\alpha)$ for all $0<\alpha < \omega_1$.
Let $K$ be the subset of $F$, of functions with
a finite range.
Question:
Is there an order-preserving homomorphism
$h : F \to K$, i.e if $f \le g$ then $h(f) \le h(g)$,
and with the additional property that $f \le h(f)$ ?
(I had dropped (b) in my first post, but comments below
show that it is essential. Also, I did mean the functions $f$ must be regressive, when I had imprecisely said decreasing, in the original statement. I added the gn tag since the question stated is an order-theoretic translation of a question from general topology: Whether $\omega_1$ has a monotone interior-preserving open operator.)
Edit April 28, 2014: The answer by Noah S below is correct but incomplete (for each $f\in F$ it finds $h(f)\in K$ with $f \le h(f)$ but does not consider whether $h(f) \le h(g)$ when $f\le g \in F$). The question is open. Thank you
Edit April 12, 2015. I reposted at MSE
Update Oct 19, 2018 (and May 21, 2019):
This questions has been included in the following paper:
Serdica Math. J. 44 (2018) (dedicated to the memory
of Professor Stoyan Nedev (1942−2015))
ON MONOTONE ORTHOCOMPACTNESS
S.G. Popvassilev, J.E. Porter
Here is a temporary link from the editors:
http://www.math.bas.bg/serdica/2018/2018-177-186.pdf
 A: OK, third time's the charm, hopefully:
First, a lemma:

Fix an arbitrary successor ordinal $\beta$ and a nondecreasing regressive map $f$ with domain $\beta-\{0\}$. Then there is a nondecreasing regressive map $d_f$ with the same domain such that $f\le d_f$ and $d_f$ takes on only finitely many values.

The proof is by induction on $\beta$. The base case $\beta=2$ is trivial, as is the successor-of-a-successor case.
Now suppose the proof holds for all non-limit $\gamma<\lambda+1$ for $\lambda$ a limit, and take $\beta=\lambda+1$. Fix some successor ordinal $\chi$ with $\chi<\lambda$ and $\chi>f(\lambda)$; such a $\chi$ must exist since $\lambda$ is limit and $f$ is regressive.
But now consider the map $F=f\upharpoonright \chi$. Since $\lambda$ is limit and $\chi<\lambda$ is a successor ordinal, we may apply the induction hypothesis to get a $d_{F}$. But now consider the map $d_f$ with domain $\lambda-\{0\}$ extending $d_{F}$ and taking on the value $max\{f(\lambda), d_F(\epsilon)\}$ (where $\epsilon+1=\chi$) on all inputs $\theta$ with $\chi<\theta \le\lambda$. This function has the desired properties - in particular, it is regressive by choice of $\chi$ - so we are done. QED

Now, to answer the actual question: fix a nondecreasing regressive $f$ defined on $\omega_1-\{0\}$. By Fodor's lemma + nondecreasingness, $f$ is eventually constant; fix some countable successor ordinal $\alpha$ such that for all $\beta>\alpha$, $f(\beta)=f(\alpha)=\gamma$.
Apply the lemma to $f\upharpoonright \alpha$ to get a function $d_0$; now extend $d_0$ to a function $d$ defined on all of $\omega_1-\{0\}$ by setting $d(\theta)=max\{f(\alpha), d_0(\epsilon)\}$ (where $\epsilon+1=\alpha$) for all $\theta\ge\alpha$.
A: The answer to the above question is no (there is no such mapping $\psi$).
The question asked above is an order-theoretic restatement of a question from general topology:
Question. Does $\omega_1$ (with its usual order topology) have a monotone interior-preserving open operator $r$ (defined below)?
Definition. A topological space $X$ in monotonically orthocompact via open refinements (abbreviated MO$_o$) if it has a monotone interior preserving open operator $r$, that is:
(i) if $\mathcal U$ is any open cover then $r(\mathcal U)$ is an interior-preserving open refinement (that covers $X$), and
(ii) if $\mathcal U$ refines $\mathcal V$ then $r(\mathcal U)$ refines $r(\mathcal V)$.
The operator $r$ will also be called an MO$_o$ operator (for $X$).
The proof that the two questions are equivalent is
in Theorem 3.1 in the following paper:
ON MONOTONE ORTHOCOMPACTNESS
S.G. Popvassilev, J.E. Porter
Serdica Math. J. 44 (2018) (dedicated to the memory
of Professor Stoyan Nedev (1942−2015))
http://www.math.bas.bg/serdica/2018/2018-177-186.pdf
As the above paper seems to be reliably available online, I will only include here a negative answer (due to Gary Gruenhage) of the topological version. Namely:
Theorem. $\omega_1$ is not MO$_o$.
This result is included in the following paper.
MONOTONE  ORTHOCOMPACTNESS  AND  PROPERTY  (A$_o$)
Gary Gruenhage, Strashimir G. Popvassilev, and John E. Porter
(accepted for publication in the journal Topology Proceedings)
The proof consists of two steps (both due to Gary Gruenhage):
(1) Every regular Hausdorff MO$_o$ space has property (A$_o$) (defined below), and
(2) $\omega_1$ does not have property (A$_o$).
I just copy these results from our paper and paste them here.
Definition 2.1. Let (A$_o$) be the following property (of a topological space $X$):
One can assign to each pair $(x,U)$, where $U$ is open and $x \in U$, an open set
$V(x,U)$ such that
(1) $x\in V(x,U)\subset U$ ;
(2) whenever $y\in \bigcap_{\alpha<\kappa} V(x_\alpha, U_\alpha)$, there is $A \subset \kappa$ such that $y \in
    \textrm{Int}(\bigcap_{\beta \in A}U_\beta)$ and for each $\alpha \in \kappa$ there is $\beta \in A$ with
$V(x_\alpha, U_\alpha) \subset U_\beta$.
Theorem 2.2. For a regular Hausdorff space $X$, MO$_o$  implies property (A$_o$).
Proof.
Let $r$ be an MO$_o$ operator for $X$. Let $U$ be open and $x \in U$. Choose open $W(x,U)$ with $x \in W(x,U)
\subset \overline{W(x,U)} \subset U$, and let $$\mathcal{O}_{x,U} = \{U, X \setminus \overline{W(x,U)}\}.$$
Then choose
$P(x,U) \in r(\mathcal{O}_{x,U})$ such that $x \in P(x,U)$ and let $V(x,U)= W(x,U) \cap P(x,U)$.
Suppose $y \in \bigcap_{\alpha<\kappa} V(x_\alpha, U_\alpha)$. Let $\mathcal{O}^*= \bigcup_{\alpha<\kappa}
\mathcal{O}_{x_\alpha, U_\alpha}$. For each $Q \in r(\mathcal{O}^*)$ with $y \in Q$, there is $O \in \mathcal{O}^*$ with
$Q \subset O$. Since $y \in V(x_\alpha, U_\alpha) \subset W(x_\alpha, U_\alpha)$ for all $\alpha$, it must be that
$O=U_\alpha$ for some $\alpha$. Choose such an $\alpha$ and denote it $\alpha(Q)$. So $Q \subset U_{\alpha(Q)}$. Finally let
$$A=\{\alpha(Q): y \in Q \in r(\mathcal{O}^*)\}.$$
We claim that
(1) $y \in \textrm{Int}(\bigcap_{\beta \in A}U_\beta)$ and
(2) for each $\alpha \in \kappa$ there is $\beta \in A$ with $V(x_\alpha, U_\alpha) \subset U_\beta.$
Note that (1) holds because each $U_\beta$ for $\beta \in A$ contains some $Q$ with $y \in Q \in r(\mathcal{O}^*)$, and
$r(\mathcal{O}^*)$ is interior preserving.
To see (2), suppose $\alpha <\kappa$. Note that
$\mathcal{O}_{x_\alpha, U_\alpha}$ refines $\mathcal{O}^*$
and so $r(\mathcal{O}_{x_\alpha,U_\alpha})$ refines $r(\mathcal{O}^*)$. Hence $P(x_\alpha,U_\alpha) \subset Q$ for some
$Q \in r(\mathcal{O}^*).$ Note that $y \in Q$ because $y \in V(x_\alpha, U_\alpha) \subset P(x_\alpha,U_\alpha).$
So now we have $$V(x_\alpha, U_\alpha) \subset P(x_\alpha, U_\alpha) \subset
 Q \subset U_{\alpha(Q)},$$ and (2) follows.
(End of proof of Theorem 2.2.}
Theorem 3.1. Let $S$ be a stationary subset of a regular uncountable cardinal $\kappa$. Then $S$ does not have property (A$_o$) (and hence is not MO$_o$; in particular $\omega_1$ is not MO$_o$).
Proof. Suppose by way of contradiction that the operator $V$ witnesses (A$_o$). Let $\alpha_0=0$. For each $x \in S$, $x>1$, we may assume
$V(x, (\alpha_0+1,x]) =(\beta_x, x]$ for some $0<\beta_x < x$. By the pressing down lemma, there is $\alpha_1 > \alpha_0$ such that
$\beta_x = \alpha_1$ for $\kappa$-many $x$ in $S$.
Similarly, there is $\alpha_2> \alpha_1$ such that $V(x, (\alpha_1 +1,x]) =(\alpha_2, x]$  for $\kappa$-many $x$. And so on.
Continue in this way to define a strictly increasing
$\kappa$-sequence $\alpha_\gamma$, $\gamma<\kappa$, of elements of
$\kappa$ such that
(i) $\forall \gamma<\kappa ( |\{x>\alpha_\gamma: V(x, (\alpha_\gamma+1,x])=(\alpha_{\gamma+1},x]\}|=\kappa$);
(ii) if $\beta$ is a limit, then $\alpha_\beta=\sup\{\alpha_\gamma: \gamma<\beta\}$.
Then $\gamma\mapsto \alpha_\gamma$ is an increasing continuous mapping of $\kappa $ onto a club subset of $\kappa$; this is also the
case if $\gamma$ is restricted to limit ordinals. It follows that there is some $\delta$ in $S$ such that $\delta=\alpha_\beta$ for
some limit ordinal $\beta$. Then $\delta=\sup\{\alpha_\gamma:
\gamma <\beta\}.$
Now we may inductively choose a strictly increasing sequence $x_\gamma$, $\gamma<\beta$, in $S$ such that
$x_0 > \delta$ and $V(x_\gamma, (\alpha_\gamma+1, x_\gamma]) = (\alpha_{\gamma+1}, x_\gamma]$.
For $\gamma<\beta$, let $U_\gamma = (\alpha_\gamma +1, x_\gamma]$ and $V_\gamma= (\alpha_{\gamma+1}, x_\gamma]$. Then $V(x_\gamma,
U_\gamma) = V_\gamma$.
Note that $\delta$ is in every $V_\gamma$ for $\gamma < \beta$.
Since $V$ witnesses property (A$_o$), there is a subset $A$ of  $\beta$ such that
(I) $\delta \in \textrm{Int}(\bigcap_{\gamma \in A}U_\gamma)$, and (II) for each $\gamma \in \beta$ there is $\eta \in A$ with
$V_\gamma \subset U_\eta$.
Since the $\alpha_\gamma$'s increase to $\alpha_\beta=\delta$, for (I) to hold the set $A$ cannot be cofinal in $\beta$. But since
the $x_\gamma$'s are strictly increasing, $A$ must be cofinal in $\beta$ for (II) to hold, so we have a contradiction.
(End of proof or Theorem 3.1)
A: I have been thinking about this question since it was bumped up. I am burned out now, so I thought I would post a few basic things that I noted. 
First, as you noted that any function $f \in \mathcal F$ would reach a constant value after some point. Also that value will obviously be the maximum value of the function. So suppose that $f:\omega_1 \rightarrow \omega_1$ maximises at a value $v_f$. Now it seems to me that this variable $v_f$ may be of some significance for the given question. 
For example, let $\mathcal F_\alpha \subset \mathcal F$ ($\alpha<\omega_1$) denote the collection of functions $f \in \mathcal F$, which satisfy the additional property the $v_f < \alpha$. Your original question was (A) Giving a function $\psi : \mathcal F \to \mathcal K$ (satisfying certain properties of course). Now I think we can also consider a sub-question: (B) Can one give a function $\psi_\alpha : \mathcal F_\alpha \to \mathcal K$ for any arbitrary $\alpha < \omega_1$ ($\psi_\alpha$ and $\mathcal K$ satisfying properties as described in question).
For example, here is an example of a function $\psi_{\omega^2} : \mathcal F_{\omega^2} \to \mathcal K$. I think, I can give a similar description for $\psi_{\omega^3} : \mathcal F_{\omega^3} \to \mathcal K$. I have not added it for the sake of brevity, but if you think the example for $\psi_{\omega^2}$ is not illustrative enough, then I will add it.  

An example for $\psi_{\omega^2}$. Given a function $f \in \mathcal F_{\omega^2}$, first determine the maximum value $v_f$ of the function. Now suppose that the maximum value $v_f$ is of the form $\omega \cdot a+b$ (with $a \in \mathbb{N}^+$, $b \in \mathbb{N}$). Denote $\psi_{\omega^2} (f)$ as $F$. Now define $F$ as follows:
$$F(x)=\omega \cdot a+b=v_f \qquad \mathrm{for} \quad  x > \omega \cdot(a+1)$$
$$F(x)=\omega \cdot a \qquad \mathrm{for} \quad \omega\cdot a < x \leq \omega \cdot (a +1)$$
$$F(x)=\omega \cdot n \qquad \mathrm{for} \quad \omega\cdot n < x \leq \omega \cdot (n+1)$$
$$F(x)=f(x) \qquad \mathrm{for} \quad x \leq \omega $$
In the third line, we have $1 \leq n < a$. (EDIT : Made a correction in the first and second equation) 
One line of thinking is to see whether we can keep giving the function $\psi_\alpha$ or not. If no, then what is the point at which we can no longer do that (the answer to (A), the question in OP, would be negative in that case). If yes, in that case the answer to (B) would be positive. Also, in that case, what would be the general requirements/obstacles at each level (to determine whether (B) implies (A) or not). Anyway, just a suggestion that might perhaps be useful (and, to be fair, possibly not useful).
EDIT2: 
Regarding the description of $\psi_{\omega^2} (f)=F$, it was rightly mentioned in comments that it is only guaranteed to work for those functions $f \in \mathcal F_{\omega^2}$ which increase smoothly. In other words, if $v_f$ is the maximum value of $f$, then $f$ satisfies the condition $\mathrm{range}(f)=v_f+1$.
Somewhat briefly, I just wanted to point out how this can be rectified somewhat easily (though it seems it would be at the cost of doubling the number of equations above). Suppose we had some function $f \in \mathcal F_{\omega^2}$ so that $f(\omega \cdot 2)=\omega+2$ and $f(\omega \cdot 3)=\omega.2+4$. Further suppose the max. value for $f$ is: $v_f=f(\omega \cdot 3)$.
Now we define $\psi_{\omega^2} (f)=F$ as follows: 
(1) For $x \leq \omega$, set $F(x)=f(x)$
(2) Set $F(\omega+1)=\omega$, $F(\omega+2)=\omega+1$ 
(3) For $\omega+3 \leq x \leq \omega \cdot 2$, set $F(x)=f(\omega \cdot 2)$ 
(4) Set $F(\omega \cdot 2+1)=\omega \cdot 2$, $F(\omega \cdot 2+2)=\omega \cdot 2+1$, $F(\omega \cdot 2+3)=\omega \cdot 2+2$, $F(\omega \cdot 2+4)=\omega \cdot 2+3$ 
(5) For $\omega \cdot 2+5 \leq x \leq \omega \cdot 3$, set $F(x)=f(\omega \cdot 3)$
(6) For $x > \omega \cdot 3$, set $F(x)=f(\omega \cdot 3)$
