The n-Category Café

The fourth or fifth thought that pops into my head is that “since” there’s exactly one two-ways of injectively monotonically mapping $[n]$ to $[n+1]$ such that the two ways are nowhere equal, there should also be (essentially) one way of mapping $\langle \mathbb{Q}, < \rangle$ to itself with no stable bounded intervals.

Hmm… I can get my head around this, though: by trichotomy etc., for any rational $x$, $\langle ]-,x[, > \rangle$ and $\langle ]x,-[,<\rangle$ are both countable infinite d.l.o.w/o.e.p.; so they’re isomorphic; choose such an isomorphism $\phi$. Then $\phi \cup \phi^{-1}\cup \{(x,x)\}$ is an order isomorphism of $\langle \mathbb{Q},<\rangle$ and $\langle \mathbb{Q},>\rangle$, fixing $x$. Similarly, there’s (far more than) one such — say $\psi$ — fixing $y$ for any $y>x$. Then $\psi\phi$ is an order isomorphism mapping $x<y$ to some $z>y$ (and now you know how I write compositions); similarly $\phi^{-1}\psi^{-1}$ maps $y$ to some $w< x$, and hence $\phi^{-1}\psi^{-1}(x)<x$. The union $U$ of the intervals $](\psi\phi)^{-n}(x),(\psi\phi)^n(x)[$ is again countable infinite dense linear without endpoints, and stable under $\psi\phi$, which acts as an order automorphism and with no stable (relatively) bounded intervals.

The argument currently playing in my head says I should look at the Euclidean algorithm to generate — from $1$, ${\cdot}+1$, and $(\cdot)^{-1}$ — all positive rationals, and then take their negatives … and then argue from some nonsense that $\mathrm{ad}_1$ as described, and any order antimorphism $\mathrm{inv}$ of $U^+= U\cap ]0,-[$ fixing $\mathrm{ad}_1(0)$ together generate a useful monoid such that the orbit of $1=\mathrm{ad}_1(0)$ is a dense order and has no endpoints. It’s then this orbit that becomes the (positive) rationals; but again it’ll be isomorphic to (half of) whatever d.l.o.w/o.e.p. you started with, by structural uniqueness.

Are you trying to get the abelian group and ring structure on $\mathbb{Q}$ to arise from its structure as a Fraissé limit? This remark of yours is obscure to me:

Could we see how to get a ⟨$\mathbb{Q}$,+⟩-torsor structure to emerge here?

If you’re trying to get a ⟨$\mathbb{Q}$,+⟩ torsor structure on something, you must already know $\mathbb{Q}$ is an additive group. But then it’s automatically torsor of itself, so trying to use Fraisse limit considerations to make some isomorphic set into a ⟨$\mathbb{Q}$,+⟩ torsor seems like an unnecessary acrobatic feat.

Of course, mathematics often profits from unnecessary acrobatic feats. But maybe I just don’t understand your starting-point and goal.

Just naively, it strikes me as strange to focus so much attention on the order structure of the rationals if you’re really trying to get to the bottom of the question “What’s so special about the rationals?” I guess it can lead to some interesting questions… but to me, and probably most mathematicians, what’s always been so special about the rationals is that they’re field of fractions of the integers.

Somehow your focus on the order structure of the rationals reminds me of this puzzle, which you may enjoy:

The rationals are a dense linearly ordered set with $\aleph_0$ points between any two distinct points. It’s an ordered field.

The real numbers are a dense linearly ordered set with $2^{\aleph_0}$ points between any two distinct points. It’s an ordered field.

Take your favorite cardinal. Is there a dense linearly ordered set with $\kappa$ points between any two distinct points? Is it unique, as a linearly ordered set? And: can you make it into an ordered field?

In other words, does the family $\mathbb{Q}, \mathbb{R}, ...$ have other members?

Speaking of Fraissé limits and the rationals, I just bumped into a curious paper entitled Towards Combining Dense Linear Order with Random Graph. I don’t understand it.

The significance of Q is that it’s the injective hull of Z. The fact that it’s a field is an incidental consequence of this more fundamental significance.