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Games are an interesting class-sized structure, with a game being a pair of sets $\\\{L\mid R \}$, which are themselves sets of games. This can be used to model finite 2 player games, but are difficult to reason about without additional structure. Two primary subFields of games exist, being the surreal numbers and the nimbers. Are there any other known subFields(or even subRings) of the games? Additionally, are there any where the unit element is not 1 or star?

This would namely be a class $F$ which is a subclass of games which is closed under Conway addition and multiplication(which I believe are definitionally commutative and are associative), and under the game equivalence, for all $a,b,c,d \in F$, if $a=b$ and $c=d$, then $a+b=c+d$ and $ab=cd$. I believe the zero of any such structure must be the canonical one, but the unit could be different, given that 1 and star are units in their fields.

I am asking this because I want to find all solutions to $x^2=x$ in the games, and any unit of a field satisfies this equation. I believe only games in simplest form need to be considered, since many games not in simplest form cause multiplication to be ill defined.

opfromthestart
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    Exactly how is multiplicaton of games defined? I see how multiplication of surreal numbers is defined but not how it extends to games. – Lucenaposition Jul 22 '24 at 22:38
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    It is defined the same way recursively, namely $xy=\{xy^L+x^Ly-x^Ly^L,xy^R+x^Ry-x^Ry^R \mid xy^L+x^Ry-x^Ry^L,xy^R+x^Ly-x^Ly^R}$. This corresponds with surreal multiplication and (I believe) nimber multiplication. I am working on finding solutions of $x*x=x$ and I believe that it can only be true for 0 or units of a field, so if there are no other subFields then those solutions are the complete set. – opfromthestart Jul 22 '24 at 22:43
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    This works for nimbers, but not for surreals such as $\uparrow$. – Lucenaposition Jul 22 '24 at 22:48
  • What do you mean that it does not work? Also, $\uparrow$ is not a surreal. – opfromthestart Jul 22 '24 at 22:49
  • I mean $\uparrow={0|}$ and $\downarrow={|0}$. If the surreals form a field, $0\times y=0$ for all $y$. Note $\times y={y^L,y^R|y^L,y^R}$ so $\times=$ and $\times\uparrow=2=*\times\downarrow$. – Lucenaposition Jul 22 '24 at 22:50
  • That would mean that multiplication does work for the game up, right? I don't understand what your point is. – opfromthestart Jul 22 '24 at 22:57
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    I agree multiplication works on nimbers, but note that it doesn't make sense to talk about subfields of the surreal games when the surreal games don't form a field. Also if $\ne0$ and $\times\uparrow=*\times\downarrow$ then $\uparrow=\downarrow$ should be true, but it is not. – Lucenaposition Jul 22 '24 at 23:01
  • I guess subfield is not the right term since games is not a field, but I meant it in the sense of having subclasses that are Fields. – opfromthestart Jul 22 '24 at 23:03
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    The multiplication definition you mentioned only works over the subsets of numbers and nimbers. See this question to see what goes wrong; in general, multiplication is not well-defined, because you can have equivalent games $G=H$ such that $G\times K\neq H\times K$, for a third game $K$. So, you still need to clarify what multiplication operations is intended over all games, because I do not know of any. – Mike Earnest Jul 22 '24 at 23:33
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    I am asking about that same multiplication operation. That link proves that $\{-1,0 \mid 0,1}$ is in no Ring, but I am asking about what other sets/classes are there other than numbers and nimbers that do respect multiplication and equivalence. – opfromthestart Jul 22 '24 at 23:51
  • It is so easy to find game values that don't play nicely with multiplication that I suspect the answer is "nothing", but I've never seen a proof nor tried to prove it myself so am upvoting. – Mark S. Jul 23 '24 at 00:28
  • Say a game is optimal if, for $x=\{L \mid R }$, $\forall l_1,l_2 \in L, l_1=l_2 \lor l_1 || l_2$, and $l_1,l_2$ are optimal, and same for $R$. Is there a triple of optimal games where $a=b$ but $ac \ne bc$? And if so, what if multiplication is also optimal, eg $a_m b$ is the optimal game where $a_m b=ab$? – opfromthestart Jul 23 '24 at 03:13
  • Your "optimal" is basically "have deleted all dominated options but we have not reverted out the reversible options, nor done those operations for any followers/proper subpositions". As such, you can't refer to "the optimal game" for your definition of $*_m$. Also, please stick to one question per post. Either change this question to be about canonical form multiplication or save that for a separate post. – Mark S. Jul 23 '24 at 10:51
  • Actually I think for respecting equality, these would be the same question I suppose. – Mark S. Jul 23 '24 at 10:56

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