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The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D). The doubling process can continue ad infinitum. -Source

Cayley Dickson

What would the $\omega$th step of this process produce?

user820789
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    So, after sedenions, we have pathions, chingons etc. How far up do we have to go to get klingons? Anyway, each embeds in the next one, so we can take a direct limit in order to get an $\omega$-th term. – Angina Seng Nov 11 '18 at 06:01
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    @LordSharktheUnknown Since you got here first, do you want to expand your comment into an answer explaining the direct limit? – Mark S. Nov 11 '18 at 19:10
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    You link a page explaining that "Triernions" don't exist, but ask if we would have them. If there is something about that page (or linked ones) you don't understand, I would recommend making that a separate question. Since the Surreals' $\varepsilon$ and $\pi$ aren't ordinals, I don't know what it would mean for something to have that many components. – Mark S. Nov 24 '18 at 00:03
  • The difficulty would be in finding a direct limit for which $|x|<\infty$ for each $x$. – John Wayland Bales Feb 03 '21 at 05:20

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As indicated in that diagram, each of these objects sits inside the bigger ones in a natural way. For instance, since octonions are pairs of quaternions, and quaternions are pairs of complex numbers, a complex number $(a,b)$ can be treated as an octonion by adding zeros: $(a,b)$ is like $(\,((a,b),(0,0)),\,((0,0),(0,0))\,)$.

The Cayley-Dickson construction is set up in such a way that the arithmetic operations (addition and multiplication) are consistent with these sorts of identifications. E.g. if you multiply two complex numbers and then view the product as an octonion, you get the same answer as if you view the complex numbers as octonions first and then multiply.

Because of these identifications, a natural step $\omega$ would be just to take the union of all of these sets of numbers, making identifications as needed. For example, when you need to test if a real number $r$ is equal to a complex number $(a,b)$, first view $r$ as $(r,0)$ and then test equality as complex numbers. Or if you need to add $(a,b)$ to $((c,d),(e,f))$, first view the complex number as the quaternion $((a,b),(0,0))$ and then add.

As Lord Shark the Unknown alluded to in a comment, whenever you have a system like this where "smaller" objects can be viewed as sitting inside larger ones by doing something (in this case: adding zeros) that's compatible with the structure (in this case: arithmetic operations), then you can collect everything together in something called the "direct limit" (in algebra). A slightly more abstract generalization of this idea would be the colimit in Category Theory.

Mark S.
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