surreal numbers

Length of surreal product

Author(s): Gonshor

\begin{conjecture} Every \Def{surreal number} has a unique sign expansion, i.e. function $f: o\rightarrow \{-, +\}$, where $o$ is some ordinal. This $o$ is the \emph{length} of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $s$ as $\ell(s)$.

It is easy to prove that

$$ \ell(s+t) \leq \ell(s)+\ell(t) $$

What about

$$ \ell(s\times t) \leq \ell(s)\times\ell(t) $$

? \end{conjecture}

Keywords: surreal numbers

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