Let $(P,\le)$ be a poset. For a point $x\in P$ let
$${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the *lower* and *upper sets* of the point $x$, and for a subset $S\subset P$, let
$${\downarrow}S=\bigcup_{s\in S}{\downarrow}s\quad\text{and}\quad{\uparrow}S=\bigcup_{s\in S}{\uparrow}s$$be the *lower* and *upper sets* of the set $S$ in $P$.

Now consider the following cardinal characteristics of $P$:

$\bullet$ the *$\downarrow$-cofinality* ${\downarrow}(P)=\min\{|C|:C\subseteq P\;\wedge \;{\downarrow}C=P\}$;

$\bullet$ the *$\uparrow$-cofinality* ${\uparrow}(P)=\min\{|C|:C\subseteq P\;\wedge\;{\uparrow}C=P\}$;

$\bullet$ the *$\uparrow\downarrow$-cofinality* ${\uparrow}{\downarrow}(P)=\min\{|C|:C\subseteq P\;\wedge \;{\uparrow\downarrow}C=P\}$;

$\bullet$ the *$\downarrow\uparrow$-cofinality* ${\downarrow}{\uparrow}(P)=\min\{|C|:C\subseteq P\;\wedge\;{\downarrow\uparrow}C=P\}$.

Proceeding in this fashion, we could define the $\downarrow\uparrow\downarrow$-cofinality ${\downarrow\uparrow\downarrow}(P)$ and $\uparrow\downarrow\uparrow$-cofinality ${\uparrow\downarrow\uparrow}(P)$ and so on.

It is clear that $$\max\{{\uparrow\downarrow}(\mathfrak P),{\downarrow\uparrow}(\mathfrak P)\}\le \min\{{\downarrow}(\mathfrak P),{\uparrow}(\mathfrak P)\}.$$

I would like to know the values of the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset $\mathfrak P$ of nontrivial finitary partitions of $\omega$.

By a *partition* I understand a cover $\mathcal P$ of $\omega=\{0,1,2,\dots\}$ by pairwise disjoint sets.

A partition $\mathcal P$ is defined to be

$\bullet$ *finitary* if $\sup_{P\in\mathcal P}|P|$ is finite (i.e., the cardinalities of the cells of the partition are upper bounded by some finite cardinal);

$\bullet$ *nontrivial* if the subfamily $\{P\in\mathcal P:|P|=1\}$ is finite (i.e., $\mathcal P$ contains infinitely many cells of cardinality $\ge 2$).

The family $\mathfrak P$ of all nontrivial finitary partitions of $\omega$ is endowed with the refinement partial order $\le$ defined by $\mathcal P_1\le\mathcal P_2$ if each cell of the partition $\mathcal P_1$ is contained in some cell of the partition $\mathcal P_2$.

It can be shown that $${\uparrow\downarrow\uparrow}(\mathfrak P)=1={\downarrow\uparrow\downarrow}(\mathfrak P),$$ so only four cofinalities (with at most two arrows) can be infinite.

Using almost disjoint families of cardinality continuum, it can be shown that ${\uparrow}(\mathfrak P)=\mathfrak c$.

Problem 1.Calculate the $\downarrow$-cofinality ${\downarrow}(\mathfrak P)$ of the poset $\mathfrak P$. In particular, is ${\downarrow}(\mathfrak P)=\mathfrak c$? Or ${\downarrow}(\mathfrak P)=\mathfrak d$?

**Remark 1.** It can be shown that ${\downarrow}(\mathfrak P)\ge\mathfrak d$.

Problem 2.Evaluate the cardinal characteristics ${\downarrow\uparrow}(\mathfrak P)$ and ${\uparrow\downarrow}(\mathfrak P)$ of the poset $\mathfrak P$.

2more comments