an infinite queue preserving equality.

Question

Is there any well-ordered set $(A,\leq)$ such that:

1. $(A,\leq^{-1})$ is well-ordered.
2. $A$ is infinite.

3. there's exactly one function $\theta:A\rightarrow \{0,1\}$ such that

   1) for each $a < M$, $$\theta(a)=\theta(a^+)$$ 2) for each $b > m$, $$\theta(b)=\theta(b^-)$$

where $$m=\min (A)$$ $$M=\max (A)$$ $$a^+=\min\{x\in A\mid a<x\}$$ $$b^-=\max\{x\in A\mid x<b\}$$

Answer 1

Even your first two requirements cannot be satisfied.

If $( A , \leq )$ is an infinite well-ordered set, then $( \mathbb{N} , \leq )$ embeds into $( A , \leq )$, that is, there is a one-to-one function $f : \mathbb{N} \to A$ such that $m \leq n \; \Rightarrow \; f(m) \leq f(n)$. Letting $X$ denote the image of this mapping it follows that $X$ has no $\leq^{-1}$-least element (since it has no $\leq$-greatest element), and thus $( A , \leq^{-1} )$ is not well-ordered.