I'm studying some topics about topological games of length $\alpha\geq\omega$, where I came across the following statement about additively indecomposable ordinals (recall that $\alpha$ is additively indecomposable if $\beta+\gamma <\alpha$ whenever $\beta,\gamma<\alpha$):


If $\alpha\geq\omega$ is additively indecomposable, then we may write $\alpha=\bigcup\_{i<cof(\alpha)} Y\_i$, where $Y\_i$ and $Y\_j$ are disjoint if $i<j$ , $Y\_i$ has order type $\alpha$ for each $i$, and $\{min(Y\_i):i<cof(\alpha)\}$ is cofinal in $\alpha$.


Although it seems to me intuitively correct (at least for countable ordinals), I could not formulate an argument to prove that such a construction is always possible. So, my question is: how can I prove the above statement?


Thank you!



A question about additively indecomposable ordinals
For the countable case, this is very easy: let $\alpha$ be countable (so $cf(\alpha)=\omega$) and additively decomposable. Then $\alpha$ is a limit, and we can decompose $\alpha$ into its "odd" and "even" bits: that is, there is a function $f: \alpha\rightarrow\alpha$ which is injective and order-preserving, such that $f(0)>0$, and $\alpha - im(f)\cong\alpha$. By iterating $f$, we get the desired partition of $\alpha$: $Y\_0=\alpha- im(f)$, $Y\_{n+1}=\alpha - (f^{n+1}(\alpha)\cup \bigcup\_{i\le n} Y\_i).$ (The condition $f(0)>0$ is used to guarantee that the minima of the $Y\_i$ are cofinal in $\alpha$.)




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For uncountable $\alpha$, I find it easier to switch to a slightly less concrete picture. Let $Y\_i=\alpha\times\{i\}$ for $0\le i<cf(\alpha)$, and let $(\gamma\_i)\_{i<cf(\alpha)}$ be an increasing cofinal subset of $\alpha$ with $\gamma\_0=0$. Let $h: cf(\alpha)^2\cong cf(\alpha)$ be a pairing function on the cofinality of $\alpha$ such that $$(\*): \,\, \text{ $\{h(0, i): i<cf(\alpha)\}$ is cofinal in $\alpha$,}$$ and for $i=h^{-1}(j, k)$ let $$L\_i=[\gamma\_j, \gamma\_{j+1}]\times\{k\}.$$ for $i<cf(\alpha)$. Then by induction, $\sum\_{i<j} L\_i<\alpha$ for each $j<\alpha$: at limit stages we use the cofinality of $\alpha$, and at successor stages we use the additive indecomposability of $\alpha$. On the other hand, we clearly have $\alpha\le\sum\_{i<cf(\alpha)}L\_i$, and so $$ \alpha\cong\sum\_{i<cf(\alpha)}L\_i.$$ This representation on $\alpha$ then pulls back to a collection of maps from the $T\_i$, which - by our condition $(\*)$ on $h$ - clearly satisfies that the images of the minima are cofinal.


