In a poset with a cofinal chain, does every cofinal subset admit a cofinal chain?

Question

Let $(P, \le)$ be a partially ordered set. A subset $A\subseteq P$ is a chain if any two of its elements are comparable. A subset $A\subseteq P$ is cofinal if every element of $P$ is less than or equal to some element of $A$.

Question: Suppose $P$ admits some cofinal chain. Does every cofinal subset $Q$ of $P$ also admit a cofinal chain?

If someone also has a reference, I'd be interested.

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Some observations that could be useful to prove this:

1. If P has a maximum element, all cofinal subsets contain the degenerate chain consisting of just that element. That's the trivial case. So we can assume P has no maximum element.

2. A possibly useful lemma (easy to see): If $P$ has a cofinal chain, then every cofinal subset of $P$ is a directed set.

3. The related question Do all cofinal chains in a partially ordered set have the same cofinality? showed that all cofinal chains in $P$ (if they exist) have the same cofinality. This question is a little different, but some similar technique may help.

Answer 1

Yes, this is true:

Let $C$ be a cofinal chain in $P$ and $Q$ a cofinal subset of $P$.

Obviously, $P$ and $Q$ are upward directed. W.l.o.g, we may assume that $\emptyset \neq C$ has no maximum, hence $C$ contains a cofinal subset order-isomorphic to a regular cardinal. Hence, we may assume that $C = \{c_\alpha: \alpha < \kappa \}$ with $\kappa$ a regular cardinal and $(c_\alpha)_{\alpha < \kappa}$ strictly increasing.

By transfinite induction we define $(q_\alpha)_{\alpha < \kappa}$ strictly increasing, such that $c_\alpha \le q_\alpha \in Q$ for all $\alpha < \kappa$.

• $\alpha = 0$: Pick $q_0 \in Q$ with $c_0 \le q_0$.
• $\alpha \rightarrow \alpha +1 $: Since $Q$ is upward directed and has no maximum, pick $q_{\alpha+1} \in Q$ with $c_{\alpha+1}, q_\alpha < q_{\alpha+1}$.
• $\lambda < \kappa, \lambda$ limit ordinal: For each $\alpha < \lambda$, choose $d_\alpha \in C$ with $q_\alpha \le d_\alpha$. Since $\kappa$ is regular, $\{d_\alpha: \alpha < \lambda\}$ has an upper bound $c \in C$. Pick $q_\lambda \in Q$ with $c, c_\lambda < q_\lambda$.

Hence, $\{q_\alpha: \alpha < \kappa\} \subset Q$ is a chain and cofinal in $P$.

I'm not aware of an explicit reference, but I'm pretty sure that this is considered to be folklore. You might find further information, perhaps including a reference, by searching for "cofinal types of directed sets". A very advanced paper is this one of S. Todorcevic. Might be one of the earliest paper on this subject, the 1940 paper of Tukey contains a reference?