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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.

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  • $\begingroup$ What's the best source of information on how to define addition, multiplication, and division directly on sign-expansions (as opposed to converting sign-expansions into LR-forms and applying Conway's recursive rules)? $\endgroup$ – James Propp Aug 10 '15 at 16:17
  • $\begingroup$ Harry Gonshor introduced sign expansions for surreal numbers (or rather, surreal numbers as sign expansions) in his Introduction to the theory of surreal numbers, but he defines field operations using LR-forms. Do you claim that every rational surreal number has an eventually periodic sign expansion? Do you know if the class of eventually periodic sign sequences is stable under field operations? If so, the field generated by $\omega$ contains only eventualy periodic sign expansions so the answer is no. $\endgroup$ – nombre Sep 28 '15 at 0:14
  • $\begingroup$ I make no such claim, because I don't know the right way to define "rational" in the context of the surreal Field or "eventually periodic" for Gonshor expansions of length > ω. (There is one trivial interpretation for which the claim holds: if "rational surreal number" means "rational real number, viewed as a surreal number under the natural embedding of the reals in the surreals".) $\endgroup$ – James Propp Sep 29 '15 at 11:46
  • $\begingroup$ Ok, I had misunderstood your post. I personnaly didn't know that rational (real) surreal numbers had eventually periodic sign expansions. I think your question is tough because computing sign expansions based on arithmetic properties is difficult. Maybe the easiest class to study is that of surreals whose Conway name contains rationals at all stages, because at least it is well defined. $\endgroup$ – nombre Sep 29 '15 at 15:35
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(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, but you can very likely also extract what you need from Philip Ehrlich's "Conway names, the simplicity hierarchy and the surreal number tree".

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  • $\begingroup$ Is this a ratio of ordinals? I don't think so, but I don't see how to prove it. Note that the presence of a minus sign is not decisive here, since for instance $\omega^2 -\omega+1$ is expressible as the ordinal $\omega^3+1$ divided by the ordinal $\omega+1$. $\endgroup$ – James Propp Nov 29 '15 at 22:22
  • $\begingroup$ Since Mark S. has answered my "two very concrete questions" (with the answers "yes" and "no", respectively), I'll approve his solution. Thanks, Mark! However, it seems to me that the original broader question is still worth exploration. It would be satisfying to have a quasi-combinatorial understanding of the sign-sequences associated with ratios of ordinals and with elements of the Field generated by ordinals. $\endgroup$ – James Propp Nov 30 '15 at 2:56
  • $\begingroup$ For instance, must the sign expansion of a ratio of ordinals have the finite-orbit property I mentioned in the original post? And must a surreal number whose sign-sequence satisfies the finite-orbit property belong to the Field generated by the ordinals? $\endgroup$ – James Propp Nov 30 '15 at 2:58
  • $\begingroup$ @JamesPropp I answered the remaining question: sign-sequences of ratios of ordinals need not have the finite-orbit property. $\endgroup$ – Mark S. Jan 16 '16 at 3:19
  • $\begingroup$ my knowledge of surreal numbers is a bit rudimentary, but I was very intrigued by your post, Mark. Quick follow up question: is it always possible to get the normal form of a surreal number expression in \omega , particularly ratios like your example, or expressions involving parentheses, powers and radicals, by approximating by a power series centered at infinity, treating \omega like a real variable? If not, is there an easy way to characterize when does or does not work? Thanks so much Rob $\endgroup$ – user92137 May 25 '16 at 20:39
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While I do not know if there is a

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

it is perhaps worth noting that Conway has identified what he believes "are perhaps the closest analogue in No of the ordinary rational numbers" (ONAG, p.47). These surreal "fractional numbers" are the continued fractions that terminate at a finite stage. Instead of using the integer part to generate the continued fraction, one uses the omnific integer part. Conway observes that: "If $x$ is fractional, so are $x+1$, $-x$ amd $1/x$ (if $x \neq 0$), but neither the sum nor the product of two fractional numbers need be fractional….” Moreover, it is easy to show that there are fractional numbers in Conway’s sense having sign-expansions that are not eventually periodic.

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