I recently came upon the following theorem which was attributed to J. Conway:


For each $A\subset \omega\_1$, let $\Phi(A)$ be a linear ordering of type $\sum\_{\alpha<\omega\_1} \tau\_\alpha$, where $\tau\_\alpha$ is $\eta$ (the order-type of the rationals) whenever $\alpha\notin A$, and $1+\eta$ whenever $\alpha\in A$. The theorem asserts that for every $\omega\_1$-like dense linear ordering is isomorphic to some $\Phi(A)$, $A\subset\omega\_1$.


My questions:


(1) Does the theorem generalize to higher cardinalities?


(2) The reference given was J. Conway's Ph.D. Thesis, but I did not manage to find anything on the web. Any help?
Thank you in advance.



Characterizing $\omega_1$-like dense linear orderings
Concerning question 2, the theorem that you mention in the case of
$\omega\_1$ is not actually difficult. I had briefly sketched a
proof of it at the conclusion of my answer to the math.SE question
[Linearly ordered sets somewhat
similar to $\mathbb{Q}$](https://math.stackexchange.com/a/174404/413), which is concerned with these types of orders---what I called $\mathcal{Q}\_A$ is the same as your $\Phi(A)$. Here is a complete argument:


**Theorem.** Every $\omega\_1$-like dense linear ordering is
isomorphic to $\Phi(A)$ for some $A\subset\omega\_1$.


Proof. Suppose that $L$ is an $\omega\_1$-like dense linear order.
Let $\langle x\_\alpha\mid
\alpha\lt\omega\_1\rangle$ be any increasing cofinal $\omega\_1$-sequence in
$L$, containing none of its limit points (i.e., scattered). Let
$\tau\_\alpha$ be the interval of points above 
$\cup\_{\beta\lt\alpha}\tau\_\beta$ and below the point $x\_\alpha$. This
is either $\eta$ or $1+\eta$, depending on whether
$\cup\_{\beta\lt\alpha}\tau\_\beta$ has a supremum in $L$ or not. These
intervals therefore realize $L$ as
$\Sigma\_{\alpha\lt\omega\_1}\tau\_\alpha$, as desired. Thus, $L$ is
$\Phi(A)$, where $A$ is the set of $\alpha$ where that supremum
exists. QED


In that previous answer, I proved that the $\Phi(A)$ orders are
determined up to isomorphism essentially by the equivalence of $A$
modulo the club filter.


**Theorem.** $\Phi(A)$ is isomorphic to $\Phi(B)$ if and only
if $A$ and $B$ agree on having $0$ and also agree modulo the club
filter, meaning that there is a closed unbounded set
$C\subset\omega\_1$ such that $A\cap C=B\cap C$. In other words,
this is if and only if $A$ and $B$ agree on $0$ and are equivalent
to $B$ in $P(\omega\_1)/\text{NS}$, as subsets modulo the
nonstationary ideal.


It would seem to be an interesting question to inquire in your
style whether one may extend this beyond $\omega\_1$ to higher
cardinals.


The anwer, unfortunately, is negative. Suppose $\kappa\gt\omega\_1$ is any cardinal. Let $A$ be the empty set, and let $B$ be any nonstationary set containing some ordinals with uncountable cofinality; for example, consider the singleton set $B=\{\ \omega\_1\ \}$. These two sets agree on a club, since both omit a club. But meanwhile, $\Phi(A)$ and $\Phi(B)$ are not
isomorphic, because the former has all points having cofinality
$\omega$, but the latter has points of uncountable
cofinality. 


