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Give an example for a linear ordered set $\langle A, ≺_A\rangle$, such that $\langle {^2}A,≺\rangle$ is not well ordered where

$$f ≺ g ↔ (∃a ∈ A)[(∀b ∈ A)(b <_A a ⇒ f(b) = g(b)) \text{ and } f(a) < g(a)].$$

My attempt: We know that $\langle A, ≤_A\rangle$ is well ordered if for every nonempty $b$ in $A$, $b$ has a minimal element with regard to $≤_A$. Hence we have to show the contrary for $\langle A2,≤\rangle$. This $A2$ (if I am not mistaking the notation) is actually $P(A)$.

However I am not sure whether I should check for another element in $P(A)$, so I am stuck on this problem. Any Help is welcome.

JMP
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    I am confusing with your notations: the standard meaning of ${^2}A$ is the set of functions from $2$ to $A$, which is naturally isomorphic with $A\times A$, which is well-ordered under the lexicographical order that you defined. However, it is like that you assume the set of functions from $A$ to $2$. – Hanul Jeon Jan 29 '21 at 11:34
  • Once you fix the notation, the key point to realise is that every infinite well-order will work here. – Asaf Karagila Jan 29 '21 at 11:50

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