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? I asked this question on math stackexchange over a week ago, but I didn't receive an answer.
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7$\begingroup$ Let me just note that the entire first-order theories are definitely not the same, since starting from quaternions the center of the algebra is $\mathbb R$ and we can express whether the algebra is $4,8,16,32,\dots$-dimensional over its center. $\endgroup$– WojowuCommented Nov 26, 2019 at 21:00
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2$\begingroup$ This paper seems relevant (but does not answer the question). $\endgroup$– Alex KruckmanCommented Nov 26, 2019 at 21:35
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$\begingroup$ If I understand the question correctly, in terms of universal algebra the question is whether the varieties of (non-associative unital) algebras generated by Cayley-Dickson algebras are all the same (from sedenions on). $\endgroup$– YCorCommented Nov 26, 2019 at 22:57
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1$\begingroup$ PS the paper referred to by Alex Kruckman is Bremner, Hentzel, Identities for algebras obtained from the Cayley-Dickson process, Comm. Algebra 29 (2001), no. 8, 3523-3534. $\endgroup$– YCorCommented Nov 26, 2019 at 23:02
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4$\begingroup$ @YCor: ``If I understand the question correctly''...This may be what is meant, but the question is written so that it asks is whether the varieties of unital semirings generated are the same, not the varieties of unital algebras. $\endgroup$– Keith KearnesCommented Nov 27, 2019 at 17:41
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