Pre-setup: Let $\kappa = \aleph_\xi$ be an uncountable regular cardinal. Its role in this question is merely to sidestep the technical difficulties surrounding the (imho quite uninteresting) notion that the surreal numbers are a proper class: instead, we will focus on the field $\mathbf{No}(\kappa)$ of surreal numbers of length (= birthdate) $<\kappa$. (It will play no important role in what follows. People who are fearless of set-theoretical difficulties may take $\kappa$ to be the class of all ordinals.)
A Construction: Since this is essential for my question, let me recall the following construction of $\mathbf{No}(\kappa)$ as an ordered field that is due to Alling and Ehrlich (references are: Alling, Foundations of Analysis over Surreal Number Fields (1987), §6.55; Ehrlich, “An alternative construction of Conway's ordered field $\mathrm{No}$”, Algebra Univ. 25 (1988) 7–16; and van den Dries & Ehrlich, “Fields of surreal numbers and exponentiation”, Fund. Math. 167 (2001) 173–188 [with erratum]).
Let $H_{\kappa}$ be the totally ordered set of functions $\kappa \to \{-1,0,+1\}$ which are eventually $0$, ordered lexicographically (with values on smaller indices having the greater weight). This set is also known as “Hausdorff's normal $\eta_\xi$-type”, and can be characterized (Harzheim, “Beiträge zur Theorie der Ordnungstypen, insbesondere der $\eta_\alpha$-Mengen”, Math. Annalen 154 (1964) 116–134, satz 10) as the unique (up to isomorphism) totally ordered set that has the $\eta_\xi$ property¹ which is the union of $\kappa$ subsets that have no subset isomorphic to $\kappa$ or to its reverse order.
Let $G_{\kappa} = \mathbb{R}(\!(w^{-H_{\kappa}})\!)_{\kappa}$ be the totally ordered additive group of formal series $\sum_{e\in H_{\kappa}} c_e\cdot w^e$ such that the support $\{e : c_e \neq 0\}$ is well-ordered for the reverse order and has ordinal $<\kappa$; such series are added pointwise, and are compared lexicographically by giving greater weight to the greatest values of $e$ (so here I am using the convention that the indeterminate $w$ is infinitely large).
Of course, at this point we cannot multiply such objects, merely add them. But we just repeat the same construction:
- Let $F_{\kappa} = \mathbb{R}(\!(w^{-G_{\kappa}})\!)_{\kappa}$ be the totally ordered field of the $\sum_{e\in G_{\kappa}} c_e\cdot w^e$ again such that the support $\{e : c_e \neq 0\}$ is well-ordered for the reverse order and has ordinal $<\kappa$; such series are added pointwise, they are multiplied as Hahn series (in $w^{-1}$), and are compared lexicographically just like at the previous step.
Then: $F_{\kappa} \cong \mathbf{No}(\kappa)$ as totally ordered fields (van den Dries and Ehrlich, op. cit., proposition 4.7(3)).
To summarize, we start with Hausdorff's normal $\eta_\xi$-type which is just a totally ordered set, we take Hahn series (of length $<\kappa$) with real coefficients a first time to get a totally ordered group, then we do it again to get a totally ordered field, and this is the field of surreals (of length $<\kappa$).
This construction is well and good, but it only gives us the totally ordered field structure on the surreals. The constructed field $F_{\kappa}$ does have a Hahn series structure, but the exponents, and the exponents of the exponents, live in a different set (resp. $G_{\kappa}$ and $H_{\kappa}$) than $F_{\kappa}$. So let us add one additional bit of construction:
Choose an arbitrary increasing bijection $\varphi\colon G_{\kappa} \buildrel\sim\over\to H_{\kappa}$. This also determines an isomorphism of totally ordered groups $\dot\varphi\colon F_{\kappa} \buildrel\sim\over\to G_{\kappa}$ by taking $\sum c_e\cdot w^e$ to $\sum c_e\cdot w^{\varphi(e)}$, and now we can identify $F_{\kappa}, G_{\kappa}, H_{\kappa}$ through the maps $\dot\varphi$, $\varphi$ and their composite. Thus we get not just a totally ordered field $F = F_{\kappa,\varphi}$, but one which is endowed with an isomorphism $F \cong \mathbb{R}(\!(w^{-F})\!)_{\kappa}$ with the set of Hahn series in $w^{-1}$ (of length $<\kappa$) with real coefficients over itself.
Note that there is considerable freedom in the choice of $\varphi$: Harzheim's theorem proceeds from a back-and-forth argument (much like Cantor's isomorphism theorem for dense unbounded linear orders, but for $\eta_\xi$ sets), so we can essentially extend any partial datum for $\varphi$ into an isomorphism.
Let us call an object obtained by this construction a surreal-like field (although, again, it's more than a field: it's a totally ordered field endowed with an isomorphism with Hahn series over itself; so, to be completely clear, a “surreal-like field” is a totally ordered $\eta_\xi$ field $F$ together with an ordered field isomorphism $\varphi$ which takes a Hahn series $\sum c_e\cdot w^e$ of length $<\kappa$ with real coefficients over $F$ to an element of $F$: in particular, $F$ has a distinguished element $w$ which is part of the datum).
In the case of Conway's $\mathbf{No}(\kappa)$, the map $\varphi$ is defined by the Conway normal form of surreals (Conway, On Numbers and Games (1976), chap. 3, p. 32; Gonshor, An Introduction to the Theory of Surreal Numbers (1986), chap. 5, esp. theorem 5.6). The indeterminate that I wrote as $w$ in the general case is $\omega$ for surreals.
Now one might be tempted to think that $\varphi$ doesn't matter here and that we always get the same structure in the end, but this is emphatically not the case: for example, in the surreal numbers there exists a unique $x$ such that $x = \omega^{-x}$ and is none such that $y = \omega^{-2y}$; but by dividing $\varphi$ by $2$ we get a surreal-like field $F_{\kappa,\varphi}$ where now $x = w^{-x}$ has no solution and $y = w^{-2y}$ has one.
(I'm sorry for the very long setup, but this is something that had been confusing me for years, including in this previous question which, in hindsight, constructs another surreal-like field that is not the field of Conway's surreals.)
So, anyway:
Question: What are some properties of $\mathbf{No}(\kappa)$ that are not shared with all other surreal-like fields?
(I am not necessarily asking for a precise characterization: we can characterize $\mathbf{No}(\kappa)$ by simply giving the rules for computing $\varphi$ on sign-extensions, as given in Gonshor's book, or we can observe some particular properties of it, like the fact that, as I mentioned, $x = w^{-x}$ has a solution but $y = w^{-2y}$ has none, but this completely fails to answer the question of what makes this particular $\varphi$ interesting. So my question should be taken to mean: how or why is $\mathbf{No}(\kappa)$ more interesting than the other surreal-like fields constructed above?)
- A totally ordered set $E$ is said to be $\eta_\xi$ whenever for any subsets $X,Y\subseteq E$ both having cardinality $<\kappa$, and such that $x<y$ for any $x\in X$ and $y\in Y$, there exists $z\in E$ such that $x<z$ for all $x\in X$ and $z<y$ for all $y\in Y$.
