Cantor's Theorem with Posets

The strengthened version is true.

Let $\langle P,\le\rangle$ be a partial order, let $2=\{0,1\}$, and let $M$ be the set of monotone maps from $P$ to $2$ with the natural pointwise partial order $\preceq$. Suppose that $\varphi:\langle P,\le\rangle\cong\langle M,\preceq\rangle$. Clearly the constant functions $\mathbf{0}$ and $\mathbf{1}$ are the minimum and maximum elements of $M$ respectively, so $P$ has a minimum element $b$ and a maximum element $t$. Let

$$f_0:P\to 2:x\mapsto\begin{cases} 1,&\text{if }x=t_0\\ 0,&\text{otherwise}\;; \end{cases}$$

then $f_0\in M$, and $f_0$ is the immediate successor of $\mathbf{0}$. Similarly,

$$g_0:P\to 2:x\mapsto\begin{cases} 0,&\text{if }x=b_0\\ 1,&\text{otherwise}\;; \end{cases}$$

is the immediate predecessor of $\mathbf{1}$ in $M$. Then $b_0=\varphi^{-1}(f_0)$ and $t_0=\varphi^{-1}(g_0)$ are the minimum and maximum elements, respectively, of $M\setminus\{\mathbf{0},\mathbf{1}\}$.

Suppose that for some ordinal $\eta$ we’ve defined $b_\xi,t_\xi\in P$ and $f_\xi,g_\xi\in M$ for $\xi<\eta$ so that for each $\xi<\eta$ we have

$$\begin{align*} b_\xi&=\min(P\setminus\{b_\zeta:\zeta<\xi\})\;,\\ t_\xi&=\max(P\setminus\{t_\zeta:\zeta<\xi\})\;,\\ \varphi(b_\xi)=f_\xi&=\min(M\setminus\{f_\zeta:\zeta<\xi\})\;,\text{ and}\\ \varphi(t_\xi)=g_\xi&=\max(M\setminus\{g_\zeta:\zeta<\xi\})\;. \end{align*}$$

Let $B_\eta=\{b_\xi:\xi<\eta\}$ and $T_\eta=\{t_\xi:\xi<\eta\}$; if $x\in P\setminus(B_\eta\cup T_\eta)$, then $b_\zeta<x<b_\xi$ for all $\zeta,\xi<\eta$. Define

$$f_\eta:P\to 2:x\mapsto\begin{cases} 1,&\text{if }x\in T_\eta\\ 0,&\text{otherwise}\;; \end{cases}$$

and

$$g_\eta:P\to 2:x\mapsto\begin{cases} 0,&\text{if }x\in B_\eta\\ 1,&\text{otherwise}\;. \end{cases}$$

Clearly $f_\xi\prec f_\eta$ and $g_\eta\prec g_\xi$ for each $\xi<\eta$. Moreover, if $h\in M\setminus\varphi[B_\eta]$, then $h(t_\xi)=1$ for each $\xi<\eta$, so $f_\eta\preceq h$, and hence $f_\eta=\min\big(M\setminus\varphi[B_\eta]\big)$. Similarly, $g_\eta=\max\big(M\setminus\varphi[T_\eta]\big)$, and we can continue to extend the recursive construction.

Suppose that at some point $P=B_\eta\cup T_\eta$. Then the function

$$h:P\to 2:x\mapsto\begin{cases} 0,&\text{if }x\in B_\eta\\ 1,&\text{if }x\in T_\eta \end{cases}$$

is monotone and not in the range of $\varphi$. Thus, the recursion defines two injections of the ordinals into $P$, which is impossible.