# set theory – Suprema of directed sets

Let $$(X, le)$$ be a partially ordered set. We call a subset $$S subseteq X$$

• … a chain if each two elements in $$S$$ are comparable with respect to $$le$$ (in other words, $$S$$ is linearly ordered with respect to $$le$$).

• directed if for all $$x,y in S$$ there exists $$z in S$$ that dominates $$x$$ and $$y$$.

Obviously, every chain is directed.

Question. Assume that every chain in $$X$$ has a supremum. Does it follow that every directed set in $$X$$ has a supremum?

Remarks.

(1) By Zorn’s lemma, $$X$$ has a maximal element (in fact, every element of $$X$$ is dominated by a maximal element of $$X$$).

(2) Since the empty set is a chain and thus has a supremum, it follows that $$X$$ has a smallest element (though this doesn’t seem to be particularly relevant to the question).

(3) Let $$D subseteq X$$ be directed. We cannot apply Zorn’s lemma directedly to $$D$$ since the supremum of a chain in $$D$$ might not be in $$D$$. What we can do is to add the set of all supremuma of subsets of $$D$$ (whenever they exist) to $$D$$, and thus obtain a new set $$tilde D$$. Then $$tilde D$$ is closed with respect to taking suprema, but I cannot see if (and why) $$tilde D$$ is directed.

Actually, the answer to the question is yes if and only if this set $$tilde D$$ is always directed: the implication “$$Rightarrow$$” is trivial, and the implication “$$Leftarrow$$” follows from applying Zorn’s lemma to $$tilde D$$ and from the fact that a maximal element in a directed set is always the supremum of this set.

(4) In general, a directed set does not necessarily contain a co-final chain. For instance, let $$mathcal{F}$$ denote the set of all finite subsets of $$mathbb{R}$$, ordered by set inclusion. Obviously, $$mathcal{F}$$ is directed; but every union of a chain of finite sets if at most countable, so $$mathcal{F}$$ does not contain a co-final chain.

(5) Let $$D subseteq X$$ be directed. We can apply Zorn’s lemma to the set $$mathcal{D}$$ of all directed subsets of $$D$$ that have a supremum in $$X$$, or to the set $$mathcal{S}$$ of all subsets of $$D$$ that have a supremum in $$X$$; so $$mathcal{D}$$ and $$mathcal{S}$$ both have a maximal element $$D_{max}$$ and $$S_{max}$$, respectively. But I see no way to show that $$D_{max}$$ or $$S_{max}$$ is co-final in $$D$$ (and thus equal to $$D$$).

(6) If $$X$$ is a lattice (i.e., every (non-empty) finite subset of $$X$$ has a supremum), then the answer to the question is yes: Indeed, let $$D subseteq X$$ be directed, and let $$mathcal{S}$$ denote the set of all subsets of $$D$$ that have a supremum in $$X$$. Then $$mathcal{S}$$ contains all finite subsets of $$D$$, and $$mathcal{S}$$ is stable with respect to monotone unions (i.e., unions of chains). This implies that $$mathcal{S}$$ equals the power set of $$D$$, so in particular, $$D in mathcal{S}$$.

Motivation. In a preprint of mine I briefly considered a similar question in the context of ordered vector spaces, and I remarked that I do not know the answer in this specific vector space setting. Now, I’m about to submit a revision of this preprint, and I noted that I do not even know the answer for general partially ordered sets (without any vector space structure).