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Let $\kappa$ be a limit ordinal endowed with its usual ordering. Given a partially ordered set $X$, a $\kappa$- ascending chain on $X$ is simply an increasing (or order-preserving, if you prefer) function $f:\kappa\to X$. We say that a $\kappa$-ascending chain $f$ stabilizes if there is an ordinal $\lambda<\kappa$ such that $f(\mu)=f(\lambda)$ for any ordinal $\mu$ with $\lambda\leq\mu<\kappa$.

I want to know if there are interesting ways to complete the following statement "For any partially ordered set $X$, every $\kappa$-ascending chain on $X$ stabilizes if and only if $\underline{\hspace{20mm}}$"?

In the case $\kappa=\omega$, we can complete the statement in the following way (assuming the axiom of dependent choice): "every nonempty subset of $X$ has a maximal element".

(The title comes from these two previous failed attempts, here and here, where you can find my motivation)

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Matemáticos Chibchas
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1 Answers1

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Since this is your third question, I will write something that I had in my mind for a while, despite the fact this is not a direct answer to your question. Think of it as a possible observation.

The reason that ascending-$\omega$-chain condition imply the existence of a maximal element is that it means that after a finite number of steps you have to stop. Finite partial orders have maximal elements. From this you can deduce something as elegant and nice as the existence of a maximal element.

On the other hand there are plenty of countable, or generally infinite, ordinals without a maximal element. This means that from an ascending $\omega_1$ chain condition you can only deduce that every well-ordered chain must be of a countable order type, but not much more. The real numbers have this condition. Every strictly increasing sequence is countable. But we can't even decide the actual cardinality of the real numbers.

I suppose that you could check into set theoretical topology cardinal invariants that the order topology of such partial order would have. Perhaps it would be something like having a basis of size $<\kappa$, or something like being $\kappa$-Lindelof. I don't know. But do note that non-linear orders have a tendency to have a crummy order topology (not always, of course).

One thing that might be interesting to see, is to see whether or not Noetherian properties translate well to this context, if the set of ideals with inclusion has ascending $\kappa$-chain condition, whether or not every ideal is $<\kappa$-generated (and vice versa); properties of modules and so on. Whether or not these would be of any use, I can't say. It is likely that for such rings to behave nicely some set theoretical assumptions on the universe would have to be added (e.g. $\lozenge_\kappa$ or something like that).

Asaf Karagila
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