This is related to this question about a "mother of all" groups, and so seemed like it'd fit in better at MO than MSE.

If I understand the answer to that question correctly, the surreal numbers have a nice characterization as being the "monster model" of the theory of ordered fields (and I think also the real-closed fields), which means that every ordered field embeds into the surreal numbers. In the answer to the question above, Joel David Hamkins gave an interesting example of what the monster model of the theory of groups would look like, which has the property that every possible group is a subgroup of this group (which caused it to be dubbed the "Hamkins' All-Encompassing Group-Like Thing," or I suppose HAEGLT, in the comments).

This question, then, is about Conway's formalization of combinatorial games, of which the surreal numbers are embedded. Conway's games are much more general than the surreal numbers, and have (among other things) the following structure:

  • There is a commutative sum of two games (which agrees with the sum on surreal numbers)
  • For any game, there is an additive inverse (so we have an abelian group)
  • There is a partial order on the games
  • There are nilpotent games, such as the star $\{*|*\}$ game of order 2, as seen in Conway's analysis of Nim

My question is, are the Conway games the monster model of the theory of... well, anything familiar related to the above? Abelian groups? Partially ordered abelian groups? Something else?

To my precise, I am sure there is probably some way to devise some artificial theory that the games are technically a monster model of. What I am wondering is if they are a monster model of some familiar algebraic theory that people use all the time, or perhaps some such theory with just a bit of added structure. Since they generalize the surreals in a fairly "natural" way, it seems intuitive that they might be a monster model of some equally "natural" theory that is more general than that of ordered fields.

EDIT: I previously wrote that the surreal multiplication can also be extended to a commutative product on the entire theory of games, as shown on (page 412 of this book). However, this is apparently not entirely true, as written in the comment below, as there is some subtlety with the equality relation.

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    $\begingroup$ In general, multiplication of games doesn't respect equality (although it does on the surreals). Indeed there can't be an associative and distributive multiplication on games with $1$ as the unit, because then we would have $0 =0\times\frac{1}{2} = (*+*)\times\frac{1}{2} = *\times(1+1)\times\frac{1}{2} = *$. $\endgroup$ – Oscar Cunningham Apr 29 at 9:22
  • $\begingroup$ Thanks, that is a good point - I will edit $\endgroup$ – Mike Battaglia Apr 29 at 9:27

In On a conjecture of Conway (Illinois J. Math. 46 (2002), no. 2, 497–506), Jacob Lurie proved Conway's conjecture that the class $G$ of games together with Conway's addition defined thereon is (up to isomorphism) the unique "universally embedding" partially ordered abelian group, i.e. for each such subgroup $A$ of $G$ whose universe is a set and any such extension $B$ of $A$, there is an isomorphism $f:B\rightarrow G$ that is an extension of the identity on $A$. The terminology "universally embedding", which is due to Conway, is unfortunate since it is sometimes confused with "universal". For partially ordered abelian groups "universally embedding" implies "universal", but I haven't checked if they are equivalent (though I suspect they're not). For ordered fields the notions are not equivalent; whereas $\mathbf{No}$ is up to isomorphism the unique "universally embedding" ordered field, it is not up to isomorphism the unique universal ordered field (though it is of course universal). I point this out in my paper Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers (J. Symbolic Logic 66 (2001), no. 3, 1231–1258). In that paper I further suggest the terminology universally extending in place of universally embedding. As an example of the potential for confusion using Conway's terminology, I point out (p. 1240) that Conway's terminology led Dales and Wooden (Super-real ordered fields, Clarendon Press, Oxford. 1996, p. 58) to mistakenly claim that $\mathbf{No}$ is up to isomorphism the unique universal ordered field.

Edit (4/30/20): For the sake of completeness it is perhaps worth adding that while Lurie's result is the deeper of the two, David Moews proved that Conway's class of games with addition (but without the order relation) is (up to isomorphism) the unique universally embedding Abelian group. See Moews's The abstract structure of the group of games in Richard J Nowakowski (ed.) More games of no chance, MSRI Publications no. 42, Cambridge University Press, Cambridge, 2002, pp. 49-57.

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    $\begingroup$ I’m glad to learn that Lurie’s early work was concrete enough for me to understand and appreciate. $\endgroup$ – Matt F. Apr 29 at 13:08
  • $\begingroup$ Category theorists would probably say that a "universally embedding" partially ordered abelian group is a partially ordered abelian group which is injective with respect to embeddings. $\endgroup$ – Tim Campion Apr 30 at 19:48
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    $\begingroup$ If I understand the statement correctly, then I think model theorists would say that (the elementary theory of) $\mathbf{No}$ is the model completion of the theory of partially ordered abelian groups. But since the theory of partially ordered abelian groups is not complete, I don't think model theorists would call $\mathbf{No}$ "the monster model" of this theory -- an incomplete theory has one monster model for each completion. $\endgroup$ – Tim Campion Apr 30 at 19:50
  • $\begingroup$ Tim Campion@. I'm not sure I understand your point. The question was not about $\mathbf{No}$ but rather about the more general class of games. Moews's result like Lurie's is about the class of games. $\endgroup$ – Philip Ehrlich Apr 30 at 20:05
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    $\begingroup$ @PhilipEhrlich I'm not questioning your answer. When I wrote $\mathbf{No}$ that was a brain-o for the class of all games. I'm just noting that the use of the term "monster model", which really goes back to the question statement, clashes with the standard usage in model theory. $\endgroup$ – Tim Campion Apr 30 at 22:36

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