Consider a Cayley-Dickson algebra $(X,+,*,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, from quaternions to octonions, we lose associativity of multiplication, and from octonions to sedenions we lose alternativity of multiplication. I conjecture that, in a sense, the sedenions are the final stop. More precisely, for any Cayley-Dickson algebra $X$ that is sedenion or beyond, is the equational theory in the signature $(+,*,0,1)$ for $X$ the same as the equational theory of the sedenions?
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1Not sure why this got a -1 ... – Noah Schweber Nov 19 '19 at 05:03
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1Now crossposted at MO https://mathoverflow.net/questions/347002/ – YCor Nov 27 '19 at 07:28
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I believe this question is unsolved. The answer I gave here cites this article which suggests that there might be an infinite sequence of identities of a certain type that are lost at each doubling step. However, this other article seems to imply that no such identity exists for $n=5$. – pregunton Dec 01 '19 at 12:06