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This is related to a previous question on How to think about ordinal exponentiation?

One possible definition for the natural product $\alpha\otimes\beta$ of ordinals is based on Cantor Normal Forms and natural sum: if $\alpha=\omega^{\alpha_1}+\cdots+\omega^{\alpha_n}$ and $\beta=\omega^{\beta_1}+\cdots+\omega^{\beta_m}$ are CNFs, then $\alpha\otimes\beta=\bigoplus_{i=1}^n\bigoplus_{j=1}^m\omega^{\alpha_i\oplus\beta_j}$.

With this definition, one automatically gets the following exponentiation law for natural products of exponentials: $\omega^\alpha\otimes\omega^\beta=\omega^{\alpha\oplus\beta}$.

My question: does it more generally hold that $\gamma^\alpha\otimes\gamma^\beta=\gamma^{\alpha\oplus\beta}$ for any $\gamma>0$? And if yes what do you recommend as a good reference?

I expected to find the answer on wikipedia page on ordinal arithmetic or on some other widely available source but did not manage.

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    $\begingroup$ I think the answer is "yes". See the bottom of p. 354 of The Cartesian product of sets and the Hessenberg natural product of ordinals by Hilbert Levitz (1979). To you or anyone else interested in ordinal arithmetic, my advice would be to get ahold of Sierpinski's book Cardinal and Ordinal Numbers. $\endgroup$ – Dave L. Renfro Nov 24 '14 at 17:37
  • $\begingroup$ The bottom of p.354 only mentions $\omega$-exponentiation. What about an arbitrary $\gamma$ (with arbitrary $\alpha$ and $\beta$) ? NB: as is standard, exponentiation is defined inductively with $\gamma^0=1$, $\gamma^{\alpha+1}=\gamma^\alpha.\gamma$, and $\gamma^{\sup_i \lambda_i}=\sup_i \gamma^{\lambda_i}$. $\endgroup$ – phs Nov 24 '14 at 20:09
  • $\begingroup$ I'll look in my copy of Sierpinski's book (at home; I'm at work now) this evening or tomorrow morning, and maybe in some other books as well, although if it's not in Sierpinski I doubt I'll find it elsewhere. I'll let you know tomorrow what I find. $\endgroup$ – Dave L. Renfro Nov 24 '14 at 20:28
  • $\begingroup$ Am I hallucinating or is the claim just wrong ?!?! Since $2=1\oplus 1$, the claim would entail $\alpha\cdot\alpha=\alpha^2=\alpha\otimes\alpha$ which does not hold in general (it holds for $\alpha=\omega$ and other simple cases). $\endgroup$ – phs Nov 28 '14 at 11:27
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I don't have an answer to your question, but I did search through quite a few set theory books this morning and I made notes of what I found in case you or others are interested.

The topic seems less covered in books than I expected, and I suspect you'll have to consult journal articles to find much of significance (unless you can read Hessenberg's and Jacobsthal's papers in their original German). To this end, a google search for all of the words Hessenberg natural sum product is the most useful search I know of for finding something if you're not able to search in a university library. I haven't had time today to do much searching for journal papers, and of the few papers I found, [9] and [10] seemed to be the most relevant, but I don't think they have anything specifically relevant to your question. The only math StackExchange post I found was When the ordinal sum equals the Hessenberg (“natural”) sum, but I didn't look very hard.

[1] Karl Heinz Bachmann, Transfinite Zahlen [Transfinite Numbers], 2nd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete #1, Springer-Verlag, 1967, viii + 228 pages.

See §23. Natürliche Operationen (pp. 107-112). The bottom of p. 109 has an identity that is what you want, but it appears to be for a natural product defined by Jacobsthal rather than the natural product as defined by Hessenberg.

[2] Abraham Adolf [Adolph] Halevi Fraenkel, Abstract Set Theory, 2nd edition, Studies in Logic and the Foundations of Mathematics, 1961, viii + 295 pages.

A 1966 3rd edition (viii + 297 pages) exists, but I don't have a copy of it. See Chapter III, §11, last two pages of Section 4. Arithmetic of Ordinals, pp. 214-215.

[3] Felix Hausdorff, Set Theory, Chelsea Publishing Company, 1957, 352 pages.

This is a translation by John R. Aumann and others of the 1935 German edition. See Chapter IV, last two pages of §14. The Combining of Ordinal Numbers, pp. 80-81.

[4] Michael Holz, Karsten Steffens, and Edmund [Edi] Weitz, Introduction to Cardinal Arithmetic, Birkhäuser Advanced Texts, Birkhäuser Verlag, 1999, viii + 304 pages.

See Chapter 1, near the end of Section 4. Arithmetic of Ordinals, p. 37. Only Hessenberg's natural sum is considered.

[5] Erich Kamke, Theory of Sets, Dover Publications, 1950, viii + 144 pages.

This is a translation by Frederick Otto Bagemihl of the 1947 2nd German edition. See Chapter IV, last two pages of §10. Polynomials in Ordinal Numbers, pp. 109-110.

[6] Azriel Levy, Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag, 1979, xiv + 391 pages.

Reprinted by Dover Publications in 2002 (xiv + 398 pages). The Dover edition includes roughly 200 Corrections and Additions in an appendix on pp. 393-398. See Chapter IV, end of Section 2. Ordinal Exponentiation, p. 130, Definition 2.21 and Exercise 2.22. Only Hessenberg's natural sum is considered.

[7] Horst Wolfram Pohlers, Proof Theory. An Introduction, Lecture Notes in Mathematics #1407, Springer-Verlag, 1989, viii + 213 pages.

A later edition exists, but I don't have a copy of it. See Chapter I, near the end of §7. Ordinal arithmetic, p. 43. Only Hessenberg's natural sum is considered.

[8] Waclaw Franciszek Sierpinski, Cardinal and Ordinal Numbers, 2nd edition revised, Monografie Matematyczne #34, PWN--Polish Scientific Publishers, 1965, 491 pages.

See Chapter XIV, Section 28: Natural sum and natural product of ordinal numbers (pp. 366-367). Despite how thorough this book is, surprisingly little is said about this topic.

[9] Philip Wilkinson Carruth, Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bulletin of the American Mathematical Society 48 #4 (April 1942), 262-271.

[10] Martin Michael Zuckerman, Natural sums of ordinals, Fundamenta Mathematicae 77 #3 (1973), 289-294.

(ADDED 31 MONTHS LATER) A few days ago I happened to come across two more items of possible interest.

[11] Rastislav Telgársky, Derivatives of Cartesian product and dispersed spaces, Colloquium Mathematicum 19 #1 (1968), 59-66.

(first few sentences of the paper) This paper contains some topological applications of Hessenberg's natural sum of ordinal numbers. Algebraic properties of this operation were studied by Sikorski in [3]. Our Theorem 1 generalizes the known formula for the derivative, i.e. the set of limit points, of a cartesian product of sets in topological spaces. Theorem 2 gives a topological definition of the natural sum and some applications to dispersed spaces. Finally, we give conditions under which the derivative of a set is closed and other related facts as well as proofs of the theorems. It seems that it [= this paper] is the first time that Hessenberg's sum [has] found an application apparently distant from its definition.

[12] Roman Sikorski, On an ordered algebraic field, Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego, Wydział III (nauk matematyczno-zycznych), Warszawa [= Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III (Sciences Mathématiques et Physiques)] 41 (1948), 69-96.

The algebraic properties of both the natural sum and the natural product seems to be mostly confined to pp. 77-78, but later pages might have some things of interest to someone only interested in the natural sum and natural product. (I didn't look very closely at the later pages.) I don't think the question that phs asked is answered in Sikorski's paper, but again, I did not spend much time looking at Sikorski's paper.

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  • $\begingroup$ Thanks a lot for all these pointers. It is always surprising to see how what seems like a simple & natural question on classic topics is not readily answered by the standard literature. $\endgroup$ – phs Nov 25 '14 at 19:20
  • $\begingroup$ "simple & natural question" -- Pun not intended? $\endgroup$ – Dave L. Renfro Nov 25 '14 at 19:24
  • $\begingroup$ Hi Dave, the results on ordinal "sums" in Carruth's paper were rediscovered later, in G.H. Toulmin. Shuffling ordinals and transfinite dimension, Proc. London Math. Soc. (3), 4, (1954), 177–195. MR0065907 (16,502a). $\endgroup$ – Andrés E. Caicedo Nov 1 '16 at 22:34
  • $\begingroup$ @Andrés E. Caicedo: FYI, I almost certainly have a copy of this paper, because I went through all the issues of this journal 7 or 8 years ago and my filter for what to photocopy was definitely weak enough to have caught this. However, those papers are still in the original folders I put them in as I was photocopying them, stacked up somewhere and not yet filed away according to topic, otherwise I would have come across it in my binders of papers on arithmetic of ordinal numbers. $\endgroup$ – Dave L. Renfro Nov 2 '16 at 14:19
  • $\begingroup$ @Dave I thought that might be the case. I only learned of Carruth's yesterday, had been previously attributing these results to Toulmin. (Once this happened, I figured you probably had something around here on the topic...) $\endgroup$ – Andrés E. Caicedo Nov 2 '16 at 15:02
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The answer is NO: it does not generally hold that $\gamma^\alpha\otimes\gamma^\beta=\gamma^{\alpha\oplus\beta}$.

For example, taking $\alpha=\beta=1$, we don't have $\gamma\otimes\gamma=\gamma^2$. Try it for $\gamma=\omega^2+\omega+1$. This gives $$\begin{aligned}\gamma^2=\gamma\cdot\gamma&=(\omega^2+\omega+1)\cdot \omega^2 + (\omega^2+\omega+1)\cdot\omega + (\omega^2+\omega+1) \\&= \omega^4+\omega^3+(\omega^2+\omega+1)\end{aligned}$$ while $\gamma\otimes\gamma =\omega^4+\omega^3\cdot 2+\omega^2\cdot 3+\omega\cdot 2+1$.

One only has $\gamma^\alpha\otimes\gamma^\beta\geq\gamma^{\alpha\oplus\beta}$ in general.

PS: It seems that the equality holds (for any exponents $\alpha$ and $\beta$) when $\gamma$ is a principal ordinal (i.e., of the form $\omega^\delta$) and even when it is a finite multiple (i.e., $\omega^\delta\cdot n$, or $\omega^\delta+\cdots+\omega^\delta$) of such an ordinal, but that goes beyond the original question.

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  • $\begingroup$ $\gamma^2=\omega^4+\omega^3+\omega^2+\omega+1$ (there are no coefficients equal to $2$). $\endgroup$ – Andrés E. Caicedo Dec 1 '14 at 19:08
  • $\begingroup$ (And principal ordinals are more commonly called (additively) indecomposable.) $\endgroup$ – Andrés E. Caicedo Dec 1 '14 at 19:09
  • $\begingroup$ ooops. Yes $(\omega^2+\omega+1)^2$ is only $\omega^4+\omega^3+\omega^2+\omega+1$. Thanks for spotting the error. One should not try to think about exponentiation before understanding multiplication. Will edit the above answer. $\endgroup$ – phs Dec 2 '14 at 7:22

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