Any references on supercomplex/surcomplex numbers?

Question

I know many text in surreal numbers have went over them but I can't seem to find anything that goes into great detail beyond a small section on them. For instance, in Foundation of Analysis over Surreal Number Fields the section on them was fairly small and not much to my liking. Is there anything that explains them in good detail?

Answer 1

$\DeclareMathOperator{\Noo}{\mathbf{No}}$I don't think any analog is known in surcomplex numbers $\Noo[i]$ to the property

"if $L < R$ are sets of surreals, then there is a simplest surreal $x$ such that $L < x < R$"

(except if one deals seperately with projections in $\Noo$ and $\Noo.i$)

No one (to my knowledge) has found a nice surcomplex exponential, nor is there a nice integration theory for surcomplex numbers (let alone a nice surcomplex analysis).

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All these notions could be worked out in surreals in order to define them for surcomplex numbers, but so far $\Noo$ has remained quite selfish in what it offers to its algebraic closure.

As a result, surcomplex numbers seem like "just" the bland universal algebraically closed field of caracteristic zero with an interesting real closed subfield. So maybe there are few texts on them because everybody's still stuck with surreals...

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(about your question, I am not sure I understand it: what does "equivalent means"?)

Answer 2

Re: your last question, certainly not, for exactly the same reason that $\mathbb{R}\not\cong\mathbb{C}$: the surreal numbers have no square root of $-1$.