This is related to [this question about a "mother of all" groups](https://mathoverflow.net/questions/126158/a-mother-of-all-groups-what-kind-of-structures-have-mother-of-alls), 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](https://books.google.com/books?id=VUVrAAAAQBAJ&pg=PA412&lpg=PA412&dq=%22conway%20product%22%20of%20games%20in%20combinatorial%20game%20theory&source=bl&ots=c9sOVd0QAk&sig=ACfU3U3fLJLj26txVgKPPS-4J0pnmHFigA&hl=en&sa=X&ved=2ahUKEwjzjrKi9IzpAhWiknIEHcHdD6QQ6AEwA3oECAoQAQ#v=onepage&q=%22conway%20product%22%20of%20games%20in%20combinatorial%20game%20theory&f=false)). However, this is apparently not entirely true, as written in the comment below, as there is some subtlety with the equality relation.



Are Conway's combinatorial games the "monster model" of any familiar theory?
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.


