Ordinal exponentiation identity with natural sum of exponents

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.

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.