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Is there a name for a partial order $\preceq$ on a set $X$ with the following property: "there exists a countable set $S \subset X$ such that for all $x \in X$ there exists $y \in S$ with $x \preceq y$"?

(Remark: If such a set $S$ exists, then it is possible to choose $S$ to be a chain: letting $(x_n)$ be an enumeration of the original (not necessarily totally ordered) set $S$, just define $\tilde{x}_1:=x_1$ and then recursively choose $\tilde{x}_n$ to be a point in $S$ that is greater than or equal to both $x_n$ and $\tilde{x}_{n-1}$.)

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  • $\begingroup$ Your remark is not entirely correct if $\preceq$ is reflexive (it is true if you consider irreflexive orders, though). Consider just a discrete countable order. Then it is dominating, but you cannot reduce it to a chain. $\endgroup$
    – Asaf Karagila ♦
    Commented Aug 8, 2015 at 22:40
  • $\begingroup$ Ah, when I wrote my remark, I was visualising $S$ as having the property that for any pair of points in $X$ there is a point in $S$ bounding both of them. Annoyingly, this means I've asked the wrong question. But I suppose that, in view of Nate Eldregde's response, I can just refer to my poset as "admitting a countable cofinal chain". $\endgroup$
    – Julian Newman
    Commented Aug 8, 2015 at 22:51
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    $\begingroup$ Yes, if your set is directed then you can get a chain bounding it. $\endgroup$
    – Asaf Karagila ♦
    Commented Aug 8, 2015 at 22:55

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Such a partial order would be said to have countable cofinality.

(The set $S$ would be said to be cofinal).

(Strictly speaking, your property could also apply to an order with finite cofinality, so if that is a possibility, then depending on what the word "countable" means to you, you might want to say "cofinality at most countable.")

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The partial orders $X$ which there is a countable increasing sequence that is cofinal in $X$ are precisely the partial orders which are $\textbf{Tukey equivalent to $\omega$}$.

Recall that a subset $D$ of a poset $X$ is cofinal if for all $x\in X$ there is some $d\in D$ with $x\leq d$ and we say that $D$ is unbounded if there is no $x\in X$ where $D\subseteq\downarrow x$. If $X,Y$ are partial ordering, then a function $f:X\rightarrow Y$ is said to be cofinal if whenever $D$ is a cofinal subset of $X$, then $f[D]$ is cofinal in $Y$. A function $g:Y\rightarrow X$ is said to be a Tukey map or an unbounded map if the image of an unbounded set in $Y$ is unbounded in $X$. We say that $X\leq_{T}Y$ ($X$ is Tukey below $Y$) if there is some cofinal mapping $f:Y\rightarrow X$. Equivalently, $X\leq_{T}Y$ if there is some Tukey mapping $g:X\rightarrow Y$. We say that $X\equiv_{T}Y$ ($X$ is Tukey equivalent to $Y$) if $X\leq_{T}Y$ and $Y\leq_{T}X$. The posets with Tukey type $\omega$ are precisely the posets $X$ with a countable cofinal chain.

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