Is ☽ a master idempotent? ☽º=☽?

Question

I found this equation in Winning Ways Volume 4:

$$☽+0=☽+*1=☽+*2=...=☽+☽=☽$$

From what I understand, this means ☽ is idempotent (I'm currently trying to learn more about loopy games & idempotents).

From what I can tell there are levels of idempotency & an idempotent could satisfy x+x=x &/or x*x=x.

The loopy idempotent games I am currently aware of are 0, on, off, dud, over & tiny.

Master idempotents seem to be the supremum of some set of idempotents ("absorbing" all others in the set). The example given in CGT - Siegel is based on multiplication (& a finite ring) & the ☽ formula is in terms of addition (& seems to include a non-finite structure?), but I think there must be some equivalence between them.

Is ☽ a master idempotent? I think the answer is clearly yes. I'm not sure if ☽ + n = ☽ holds for all values of n (like $\uparrow$, tiny, on, etc.) or if there any exceptions. Additionally, does ☽ satisfy $☽n=☽$? $☽^☽=☽$? I suppose my underlying question is more "Is ☽ a type of master master idempotent? / Is there a supremum of master idempotents?"

Is $☽^{o}=☽$? My guess is yes, but I'm not 100% certain.

I'm also not sure if equations like $☽=\{☽\}$ & $☽=\{☽|☽\}$ are valid for ☽ or not.

Answer 1

General Advice

(This section doesn't directly address your questions, but I think it's relevant to their source.)

1. If you take only one thing away from this answer, I would emphasize that: across all of mathematics, an equation or statement depends very sensitively on context. For example, "For every $x$, $x^2+1\ne0$." is true in the context of the real numbers and false in the context of the complex numbers. Mixing or combining contexts must be done with great care and understanding.
2. Siegel's book "Combinatorial Game Theory" (CGT) is a graduate-level text. If you haven't already studied standard mathematical curricula to the point where, say, you've learned what "supremum" means in real analysis or order theory, I would recommend not jumping into Siegel yet.
3. "Winning Ways for Your Mathematical Plays" (WW) is very challenging for self study, because it covers an amazingly broad array of separate contexts, makes many claims without justification, generally doesn't provide exercises to confirm/reinforce your understanding, etc. For an introduction to combinatorial game theory, I would recommend starting by following a standard undergraduate text, like "Lessons in Play: An Introduction to Combinatorial Game Theory" by Albert, Nowakowski, and Wolfe or "An Introduction to Combinatorial Game Theory" by L. R. Haff & W. J. Garner. Working through one of those textbooks in order would give you a solid foundation for further study/questions online, etc.

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Contexts

The question talks about mathematical objects and notations in a wide variety of (partially-overlapping) contexts, and it will be important to keep them clear, so I will do my best to list them here. I will follow Siegel in using "game" to refer to a game position like $*3$ (as opposed to a "ruleset" like Nim).

1. The context of impartial games in not-quite-disjunctive sums (confusingly also written with "$+$") where some moves are designated as "entailing moves" which require a response to be in the same component of the sum. In this context, winning moves in a sum (under normal play) can be determined by assigning sets of nimber values to each game (which effectively give information about their equivalence class under this "sum") and using nimber addition on the relevant values (which depend on the entailing moves). This is the context where $☽+0=☽+*1=☽+*2=\cdots=☽+☽=☽$ appears in WW.
2. The context of (possibly loopy) partizan games in which $+$ represents genuine disjunctive sums, and notation like $\{\text{blah}\mid\text{bloo}\}$ is useful to encode a game by listing the moves for the players Left and Right. In this context, the "degree of loopiness" of a game $G$ is written $G^\circ$ and it is convenient when $G^\circ$ is idempotent under disjunctive sum.
3. The context of sets of impartial games that are closed under disjunctive sum and making moves and played with the misère play convention. This is the context in which Siegel found it useful to define the "master idempotent" in order to develop the theory of kernels of finite misère quotients.
4. Contexts of certain numbers, in which $x^x$ is meaningful. Note that even positive infinite surreals do not have a single default definition of binary exponentiation to allow this.

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Questions and Answers

Quick Answers

Idempotence of ☽

this means ☽ is idempotent

It is true that $☽+☽=☽$ means that the equivalence class represented by $☽$ is idempotent under the non-quite-disjunctive sum that $+$ signifies in context #1.

I'm currently trying to learn more about loopy games & idempotents

Since context #1 has no loopy games, any context-#2 discussion of idempotent loopy games is completely unrelated to this idempotence of $☽$.

Loopy idempotents

The loopy idempotent games I am currently aware of are 0, on, off, dud, over & tiny.

Yes, those are all idempotent (possibly-)loopy games. $\spadesuit$ is another, I believe.

Since context #2 has no entailing moves (and has partizan games), these are in no way related to $☽$.

Degree

Is $☽^\circ=☽$?

The short answer is that $☽$ doesn't arise in the context of loopy games, so how could it have a degree of loopiness?

More precisely, since $G^\circ$ is defined in terms of an upsum, and $☽$ doesn't represent anything partizan, I'd say $☽^\circ$ isn't even defined. (And if you were to say that upsum should just be disjunctive sum in the nonloopy case and that the negative $\overline{☽}$ is just $☽$ in the impartial case, I'd still say it's undefined, because games with entailing moves aren't really using disjunctive sum.)

Adding $n$

I'm not sure if $☽ + n = ☽$ holds for all values of $n$ (like $\uparrow$, tiny, on, etc.)

With the debatable exception of $n=*m$ for some nonnegative integer $m$, the expression $☽ + n$ isn't even defined. This is because $☽$ in context #1 is basically an equivalence class of impartial games with special entailing moves where $+$ doesn't have its usual disjunctive-sum-related meaning(s), and things like $\uparrow$, $\mathbf{tiny}$, and $\mathbf{on}$ (context #2) are partizan games (or equivalence classes thereof under disjunctive sum) with no contextual concept of entailing moves.

Other operations

does $☽$ satisfy $☽n=☽$? $☽^☽=☽$?

Multiplication is typically only defined for nimbers or surreal numbers, so $☽n$ isn't defined for any $n$. (If you meant to take things like $☽4$ to mean $☽+☽+☽+☽$, then sure, under that iterated addition meaning, for positive integer $n$ only, we have $☽n=☽$.)

Arbitrary binary exponentiation is typically only defined for...I guess positive integer( surreal)s? Maybe positive rational( surreal)s? In any case, $☽^☽$ is not even close to being defined.

Set notations

I'm also not sure if equations like $☽=\{☽\}$ & $☽=\{☽\mid☽\}$ are valid for $☽$ or not.

There are many things you might mean by this notation, but they either aren't defined, or at best, are false.

Usually, when $☽$ is thought of as a set, it's a set of nimbers in context #1. Viewed as a set that way, $☽=\varnothing$. So it can't be $\{☽\}$. Worse still, since $☽$ isn't a nimber, $\{☽\}$ isn't a set among the sets of nimbers that help us analyze games with entailing moves in context #1.

However, if you meant for $\{☽\}$ to suggest something like "a game where all moves are to a position with value $☽$" (you'd have to explain that that's how you're using the set notation) then $\{☽\}$ makes some sense, but would not be equal to $☽$ since maybe someone could win by not moving in that component.

For $\{☽\mid☽\}$, things are similar to the above discussion, but worse in the sense that I don't think anyone has defined partizan games with entailment, so this would seem to be meaningless.

Master Idempotents

What is a (master) idempotent?

an idempotent could satisfy x+x=x &/or x*x=x.

Sure. Whatever binary operation you're looking at, a thing $x$ could be idempotent if when you do the operation with $x$ and $x$, you get $x$ as the result.

Making the obvious generalization beyond context #3, Siegel defined a "master idempotent" (albeit in scare quotes since this was not a term he used more than once) of a finite commutative semigroup to be the product/sum of all of the idempotents.

From what I can tell there are levels of idempotency

I suppose this is a matter of opinion, but I'd say there aren't; something either is idempotent or it isn't. I expect that Siegel didn't intend to convey that the master idempotent was somehow more idempotent than the other idempotents, just that it was kind of a "master" of them in that if $z$ is the master idempotent and $x$ is an idempotent, then $zx=z$. To quote the OP, "'absorbing' all others in the set".

Is $☽$ a type of master master idempotent?

I honestly don't know what this is intended to mean. A master idempotent is necessarily unique.

Master idempotents in other contexts?

I think there must be some equivalence between them. The way the operation is written doesn't really matter, though I will point out that context #3 where Siegel defined master idempotents is neither context #1 nor #2.

In context #2, while there aren't only finitely many idempotents so that the above definition doesn't apply as-written, one might argue that $\mathbf{dud}$ is the master idempotent since $\mathbf{dud}+x=\mathbf{dud}$ for any game $x$.

Is $☽$ a master idempotent? I think the answer is clearly yes.

In context #1, I don't think it's immediately clear. I'd have to think through how games and their values work in this context. But I expect that $☽$ could very well be the master idempotent.

Suprema

It's not clear to me how supremum is being used in the OP. It's just a least upper bound under a relevant ordering. The supremum of $\{1,3,5\}$ is $5$.

Master idempotents seem to be the supremum of some set of idempotents .

Under the ad-hoc partial order where $x\le y$ if and only if $xy=x$, then yes, a master idempotent is the maximum and hence the supremum of all of the idempotents.

Under the standard ordering of games in context #2 by favorability to the player Left, the master idempotent $\mathbf{dud}$ would not seem to be a supremum of any useful set of idempotents, since the fundmental idempotent $\mathbf{on}$ is not less than $\mathbf{dud}$.

Is there a supremum of master idempotents?"

A master idempotent is necessarily unique, so the supremum of the singleton set of master idempotents is the one master idempotent, I guess.