Three Dollar Quill

All right, apparently the secret to getting stuff done was to have no expectations about what or whether I was going to work on. Shortly-ish after posting the last entry, I worked out that a hopefully-equivalent means of comparison to the stuff I was talking about at the end there was to define comparators that involve an ordered pair of distinct indices. For surreal numbers, this means there are two relations, one for each order. For these attempted surcomplex numbers, there are six relations. I'm choosing to define the relations so that if all relations hold for a pair of numbers, they're considered equal. And to determine whether a triple actually corresponds to a number, we need to take three of these relations, and apply them to the sets corresponding to the relation's indices; if none of the pairs of members from two sets satisfy the corresponding relation, then the triple is a number.

That's the conjecture, anyway. I haven't put in the legwork to verify it yet, but first, a few more implications:

• Surreal numbers have two relations with one combination not corresponding to a number.
• Standard surcomplex numbers should thus have seven combinations not corresponding, I think.
• These things have six relations, with a whopping 51 combinations not corresponding to a number.
• Not sure if those are all accessible.
• These are boolean relations, so each of the six possible turn orders should have one of two outcomes; because each relation corresponds to a pair of sets (players), that means that I believe modeling the game version as two players assigned two of three sides makes the most sense.
• I'm unsure how to model this in the Hackenbush style; I think I'll just punt on that.
• If these are two-player games, then the nimbers are going to act very similar to the surreal version.
• However, there seem to me to be some sort of "quasi-nimber", which is equivalent to zero on some axes, but not others.
• Add the three simplest quasi-nimbers up to get... the ordinary nimber?
• But wait, if that's three, what about two?
• Hm. It seems there are three axes to consider, resulting in a four-tier scale of... nimerality? Nimericness?

One question I have is, as far as representing this stuff in code, how do I want to distinguish between structural equality and numerical equality? For now, I'm going to sleep on this.

Good night.