What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?


I can think of more than one natural way to generalize the notion of a rational number, but "ratio of omnific integers" is not one of them, since every real number is a ratio of omnific integers. Maybe ratios of ordinals are the thing to look at (as a generalization of the non-negative rationals). Or maybe we should look at the Field-closure of the ordinals. Or perhaps we should consider the Class containing every surreal number whose normal form involves only rational numbers at all levels. 


I can also think of more than one way to generalize the notion of eventual periodicity to sign-sequences indexed by a general ordinal alpha. One of them is a variant of "Kaufman decimals" (see <https://mchouza.wordpress.com/2013/08/25/kaufman-decimals/> and <http://www.jefftk.com/p/decimal-inconsistency>) in which the digit-set {0,...,9} is replaced by {$+$,$-$} and every over-bar is assigned an ordinal.


A seemingly different but possibly equivalent notion generalizing eventual periodicity involves a kind of symbolic dynamics I haven't seen before, where the Monoid of ordinals acts on ordinal-indexed sequences: if $s$ is a sequence indexed by some initial segment of the ordinals, and $\iota$ is some ordinal, define $T^{\iota}(s)$ to be the sequence obtained by omitting the first $\iota$ terms of $s$ (with $T^{\iota}(s)$ defined to be the empty sequence if $\iota$ is greater than or equal to the length of $s$). Then eventual periodicity (in the case where $s$ is indexed by the natural numbers) is seen to be a special case of the condition that the orbit of $s$ under the action of the Monoid of all ordinals is finite. (See Joel Hamkins' recent post, showing that constant sequences satisfy this finiteness condition: <http://jdh.hamkins.org/every-ordinal-has-only-finitely-many-order-types-for-its-final-segments/>.)


If my original question seems too vague (what does "right generalization" mean?), here are two very concrete ones that are relevant: does the surreal number with sign-expansion $+-^{\omega}++-^{\omega}+++-^{\omega}++++-^{\omega}\cdots$ (indexed by $\omega^2$) lie in the field generated by $\omega$? and, is it expressible as a ratio of ordinals? (Note that this sign-sequence does not satisfy the aforementioned finiteness property, although perhaps it satisfies some weaker regularity condition.) Here $-^{\omega}$ denotes a string of $\omega$-many $-$'s and $\cdots$ denotes that the pattern continues $\omega$ times.


I'd be interested in any implications that might hold between these various properties. 



Nice sign-expansions of special surreal numbers
(This answer has been substantially edited since its original version.)


### 1.


The surreal number with sign-expansion $+-^{\omega}++-^{\omega}+++-^{\omega}++++-^{\omega}\cdots$ (indexed by $\omega^2$) can be rewritten in a tidy form, which shows that it *does* lie in the field generated by $\omega$.


As you read off that sign-expansion, the number follows the following pattern: $$1,1/2,1/4,1/8,\ldots$$
$$+-^{\omega}=\frac1\omega$$
$$\frac2\omega,\frac3\omega,\frac2\omega+\frac1{2\omega},\frac2\omega+\frac1{4\omega},\ldots$$
$$+-^{\omega}++-^{\omega}=\frac2\omega+\frac1{\omega^2}$$
$$+-^{\omega}++-^{\omega}+++-^{\omega}=\frac2\omega+\frac3{\omega^2}+\frac1{\omega^3}$$


So that we end up with $$\frac2\omega+\frac3{\omega^2}+\frac4{\omega^3}+\cdots=\frac{2\omega-1}{\left(\omega-1\right)^2}$$
Now, the fact that the finite sums and the subtler fact that the "infinite sum" works out in this way is not obvious.


### 2.


$\dfrac{2\omega-1}{\left(\omega-1\right)^2}$ is not actually a ratio of ordinals.


Treating the expression as a rational function of a real variable $\omega$, it diverges at $\omega=1$, but if it could be written as a rational function with positive coefficients (say by multiplying the numerator and denominator by a suitable polynomial), it could not diverge for any positive $\omega$. Incidentally, the inverse of this sort of argument is in Meissner's "Über positive Darstellungen von Polynomen" which contains the theorem that a polynomial which is positive for nonnegative real $x$ (such as $x^2-x+1$) can always be written as a rational function with positive coefficients (such as $(x^3+1)/(x+1)$).


### 3.


A sign expansion with the finite-orbit property need not be in the field generated by the ordinals.


$+^\omega-^{\omega^2}$ has the finite orbit property (indeed, the only nontrivial tails have the forms $+^\omega-^{\omega^2}$, $-^{\omega^2}$), but $+^\omega-^{\omega^2}=\sqrt{\omega}$ is not in the field generated by the ordinals (and hence certainly not a ratio of ordinals).


### 4.


A ratio of ordinals that are not both finite can still sometimes have the finite-orbit property. In particular, $\dfrac{1}{\omega}=+-^{\omega}$, and more interestingly, $\dfrac{1}{\omega+1}=\left(+-^{\omega}-+^{\omega}\right)^{\omega}$. The latter expansion has the finite orbit property, since the only forms of tails are $\left(+-^{\omega}-+^{\omega}\right)^{\omega}$, $-^{\omega}-+^{\omega}\left(+-^{\omega}-+^{\omega}\right)^{\omega}$, $-+^{\omega}\left(+-^{\omega}-+^{\omega}\right)^{\omega}$, and $+^{\omega}\left(+-^{\omega}-+^{\omega}\right)^{\omega}$.


$+-^{\omega}-=\dfrac{1}{\omega}-\dfrac{1}{\omega2}=\dfrac{1}{\omega2}$ Then for finite $n$, we have $+-^{\omega}-+^{n}=\dfrac{1}{\omega}-\dfrac{1}{\omega2}+\dfrac{1}{\omega4}+\cdots+\dfrac{1}{\omega2^{n-1}}=\dfrac{1}{\omega}-\dfrac{1}{\omega2^{n+1}}$. We end up with $+-^{\omega}-+^{\omega}=\dfrac{1}{\omega}-\dfrac{1}{\omega^2}$. Analogously, $\left(+-^{\omega}-+^{\omega}\right)^n=\displaystyle\sum\_{k=1}^{2n}\dfrac{(-1)^{k-1}}{\omega^k}$. By the expansion of $\dfrac{1}{x+1}$ at infinity, we have $\dfrac{1}{\omega+1}=\left(+-^{\omega}-+^{\omega}\right)^\omega$.


### 5.


However, a ratio of ordinals need not have the finite-orbit property. In particular, $\dfrac{1}{\omega+2}=+-^{\omega}-+^{\omega}-^1+-^{\omega}+^3-+^\omega-^7+-^\omega+^{15}\cdots$. In a similar way to the analyses of $\dfrac{2\omega-1}{(\omega-1)^2}$ and $\dfrac{1}{\omega+1}$, this follows from the expansion at infinity: $\dfrac{1}{\omega+2}=\displaystyle\sum\_{k=1}^{\infty}\dfrac{(-2)^{k-1}}{\omega^k}$.


Enough tools for justifying the above facts about sign expansions above are covered in VIII.2 and VIII.3 from Aaron N. Siegel's book [Combinatorial Game Theory](http://www.ams.org/bookstore-getitem/item=gsm-146), but you can very likely also extract what you need from Philip Ehrlich's "[Conway names, the simplicity hierarchy and the surreal number tree](http://www.ohio.edu/people/ehrlich/ConwayNames.pdf)".


