Can a partially ordered set always be partitioned by its chains?

Question

I would like any references that discuss how to partition a partially ordered set into subsets such that each subset is a chain in the partial order. For example, I am thinking that any partial order (or at least a bounded lattice) can be partitioned in such a way that the sets comprising the partition are chains of the partial order, but I have no references that discuss this problem when the partial is infinite and not the Boolean lattice. Thank you.

Answer 1

Yes, the transfinite induction in your comment works!

Let $f$ be a function that maps every nonempty subset of $P$ to a maximal chain in that subset (we get $f$ using Choice).

Now we use transfinite recursion to define a sequence of subsets of $P$:

• $S_0 = P$
• If $S_\alpha = \emptyset$, then $S_{\alpha + 1} = \emptyset$. Otherwise, we have $S_{\alpha + 1} = S_\alpha - f(S_\alpha)$.
• For every limit ordinal $\alpha$ we have $S_\alpha = \bigcap_{\beta < \alpha} S_\beta$.

Let $S = \bigcap_{\alpha} S_\alpha$. Now, we have some smallest ordinal $\gamma$ such that $S_\gamma = S$: For every element $x$ of $P - S$, let $\gamma_x$ be the smallest ordinal such that $x \notin S_{\gamma_x}$, and then let $\gamma$ be the limit ordinal of the set $\{\gamma_x : x \in P - S\}$.

Now, we know that $S = \emptyset$, since otherwise, we would have $S \subseteq S_{\gamma+1} \subset S_\gamma = S$.

So the set $\{f(S_\alpha) : \alpha < \gamma\}$ will be a partition! This is because for some $\alpha < \beta$, we have $S_\beta \subseteq S_\alpha - f(S_\alpha)$, so the ses $f(S_\alpha)$ and $f(S_\beta)$ are disjoint, and because $\bigcup_{\alpha < \gamma} f(S_\alpha) = P - S_\gamma = P$.