Sign-expansion definition of Surreal arithmetical operations

Question

Is there a way to define the addition and multiplication operations in Surreals numbers, defined directly on the sign-expansion notation {-,+}, i.e. without firstly convert them to the Conway notation {L|R}?

Answer 1

A long time ago, I convinced myself that the following procedure has the effect of adding one to a surreal number directly on the sign expansion:

• repeatedly (and transfinitely) skip blocks of identical signs whose length is an ordinal multiple of $\omega$,

• at this point (the next block of identical signs is finite):

  • if the end of the sequence has been reached, or if the next sign is a $+$, insert a $+$ sign, otherwise [the next sign is $-$]

  • if there follow at least two (but necessarily finitely many!) consecutive $-$ signs, remove one of them, otherwise [there is a single $-$ sign]

  • if there follows a single minus sign which is the last sign in the sequence, remove it, otherwise [there is a single $-$ sign followed by a $+$ sign]

  • replace the ${-}{+}$ sequence (which is known to follow) by ${+}{-}$.

(I'm didn't check this too carefully, so maybe there are mistakes, but if this is wrong the correct procedure should be at least similar to this.)

This is not very pleasant! So it doesn't seem likely that one can describe the general case of addition, let alone multiplication, in any way that is remotely useful.