## Lectures by Aaron Siegel at the Weizmann Institute

Aaron Siegel has come back from Israel where he gave five two-hour lectures on misere games.

- Normal play
- Octal games and misere play
- The periodicity theorem
- More examples
- Further topics

Students and faculty from the Weizmann helped Aaron prepare excellent notes that are now available (PDF).

Many thanks to Aviezri Fraenkel at the Weizmann Institute for creating this opportunity to spread the word about misere games and their intriguing algebra!

## Misere quotients for impartial games

Elwyn Berlekamp suggested that the title of my recent paper with Aaron Siegel,* The Phi-values of various games*, be changed:

I urge you to change the title. I think you have a good chance to make an impact well beyond the small circle of experts such as those who come to conferences like the one in Banff. There are many more mathematicians and algebraists out there, as well as a big computer Go community. Terms like “Phi-function” are unlikely to become buzzwords that stick, as there are many Phi functions defined by many different authors. “Misere Quotient” might well be a unique pair of words. It could quite plausibly become buzzword that would help stir up more interest in combinatorial games in general. So I think it important that your title include that pair of words. I’d propose a short title such as “Misere Quotients for Impartial Games”.

It’s hard to argue with that, and it’s still a preprint. So Misere Quotients for Impartial Games it is!

## Misere solutions for particular octal games

Some commutative monoids for specific games are now online for you to browse, if you like.

There’s more to come!

## The commutative algebra of T2

I’ve been reading about semigroup rings and other things in commutative algebra. In looking though old notes I found this email from Aaron Siegel from about a year ago, recounting some interesting information from a meeting with David Eisenbud. It corresponds pretty closely to what I was trying to compute myself, this morning

The meeting went well. He showed me a new (to me) way of looking at monoids in terms of commutative rings. Namely, instead of considering the monoid

Q = {a,b : a^2=1,b^3=b }

[ Aside: the monoid Q is the (tame) misere quotient of the game of Nim played using heaps of size 1 and 2 only. In this paper, Q is called T_{2}. ]

we instead look at the ring

R = k[a,b] / (1-a^2,b-b^3)

and now we can apply the techniques of commutative algebra to analyze this ring. One observation is that because Q six elements, R is a 6-dimensional vector space over k. A basis for this vector space is Q itself, but it’s instructive to look at other bases and to study various decompositions of R.

For example, if x is an idempotent of R, then so is 1-x, since (1-x)(1-x) = 1-2x+x^2 = 1-2x+x = 1-x. Further, x(1-x) = 0, so R can be written as the internal direct sum of R/(x) and R/(1-x), with the isomorphism

R -> R/(x) + R/(1-x)

given by y -> ((1-x)y,xy).

Using the above example, we have

R = R/(b^2) + R/(1-b^2),

and it’s not hard to see that R/(1-b^2) is four-dimensional with basis {b^2,ab^2,b,ab} (which you’ll recognize as a maximal subgroup of Q), and R/(b^2) is two-dimensional with basis {1-b^2,a-ab^2} (which of course is equal to {1,a} in R/(b^2)).

Note the parallel with semigroup theory. In the ring R/(1-b^2), we’ve asserted that b^2 = 1, making b^2 into the identity. So the group {b^2,ab^2,b,ab} corresponds exactly with the multiplicative structure of R/(1-b^2). In other words, R/(1-b^2) is isomorphic to the group ring k[Z_2 x Z_2]. Likewise, in R/(b^2) we’ve set b^2 = 0 and the result is isomorphic to the group ring k[Z_2], corresponding to the maximal subgroup {1,a}. So the ring decomposition by idempotents matches the monoid decomposition we’re used to (archimedean components).

All very interesting. Eisenbud also pointed out that the rings can be decomposed further:

1 – b^2 = (1-b)(1+b) so R/(1-b^2) = R/(1-b) + R/(1+b),

and we can further break off the “a” part of each component, since it never interacts with the “b” component. This fully decomposes R as the vector space product of six copies of k and gives a new basis.

Finally, we know that we always have a^2 = 1 in any misere quotient Q. Because of this, the corresponding ring R always decomposes into two isomorphic components

R = R/(1-a) + R/(1+a)

and it’s simpler, therefore, to discard a and study just one of these (R/(1-a) is preferable, since that’s equivalent to just setting a = 1.) Of course, the structure of the P-positions is different on each component. It’s unclear what all of this might mean in game-theoretic terms.

He also recommended a book by Gilmer, “Commutative Semigroup Rings,” which I’ve checked out from Berkeley’s library. The paper by Eisenbud & Sturmfelds, “Binomial Ideals,” might also be useful (it’s on the arxiv).