I've to look deeper on this power associativity... sounds familiar...
with unitBtw I agree. Basically if we regard as hypercomplex everything that is a finite dimensional algebra, not necessarily commutative or associative, over the real, e.g. (C, H, O, S) , in the last years, doing some readings for hobby... I grow more convinced that those structures conceal a great deal of fundamentally natural properties that has to do with the very essence of the universe in some way.
For example I tried, not very successfully, to follow John Baez on his blog posts about the octonions and their relevance to the standard model, or the dimensions in which some kind of hypercomplex algebras has some properties that have to do with the dimensions were spacetime appears in some obscure and esoteric string theory black magic, or still about dual numbers were we can rebuild analysis algebraically by the use of idempotent elements acting as infinitesimals.
EDIT: I'm not sure that my brain can process that 8-dim caley table. I might try too look at it with more calm. I'm way less expert at it than you btw.
with unitBtw I agree. Basically if we regard as hypercomplex everything that is a finite dimensional algebra, not necessarily commutative or associative, over the real, e.g. (C, H, O, S) , in the last years, doing some readings for hobby... I grow more convinced that those structures conceal a great deal of fundamentally natural properties that has to do with the very essence of the universe in some way.
For example I tried, not very successfully, to follow John Baez on his blog posts about the octonions and their relevance to the standard model, or the dimensions in which some kind of hypercomplex algebras has some properties that have to do with the dimensions were spacetime appears in some obscure and esoteric string theory black magic, or still about dual numbers were we can rebuild analysis algebraically by the use of idempotent elements acting as infinitesimals.
EDIT: I'm not sure that my brain can process that 8-dim caley table. I might try too look at it with more calm. I'm way less expert at it than you btw.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)