Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset of nontrivial finitary partitions of $\omega$

Question

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 $$\max\{{\uparrow\downarrow}(\mathfrak P),{\downarrow\uparrow}(\mathfrak P)\}\le \min\{{\downarrow}(\mathfrak P),{\uparrow}(\mathfrak P)\}.$$

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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 $\mathcal P_2$.

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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 ${\uparrow}(\mathfrak P)=\mathfrak c$.

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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 ${\downarrow}(\mathfrak P)\ge\mathfrak d$.

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

Answer 1

At the moment we have the following information on the cofinalities of the poset $\mathfrak P$ (see Theorem 7.1 in this preprint).

Theorem.

1) ${\downarrow}\!{\uparrow}\!{\downarrow}(\mathfrak P)={\uparrow}\!{\downarrow}\!{\uparrow}(\mathfrak P)=1$.

2) ${\downarrow}(\mathfrak P)={\uparrow}(\mathfrak P)=\mathfrak c$.

3) ${\downarrow}\!{\uparrow}(\mathfrak P)\ge \mathrm{cov}(\mathcal M)$.

4) $\mathsf \Sigma\le{\uparrow}\!{\downarrow}(\mathfrak P)\le\mathrm{non}(\mathcal M)$.

Here $\mathrm{non}(\mathcal M)$ is the smallest cardinality of a nonmeager set in the real line, and

$\mathsf \Sigma$ is the smallest cardinality of a subset $H$ in the permutation group $S_\omega$ of $\omega$ such that for any infinite sets $A,B\subseteq \omega$ there exists a permutation $h\in H$ such that $h(A)\cap B$ is infinite.

By Theorem 3.2 in this preprint, $$\max\{\mathfrak b,\mathfrak s,\mathrm{cov}(\mathcal N)\}\le\mathsf\Sigma\le\mathrm{non}(\mathcal M).$$ The cardinal $\mathsf\Sigma$ is equal to the cardinal $\mathfrak j_{2:2}$, discussed in this MO-post.

However I do not know the answer to the following

Problem. Is ${\downarrow\!\uparrow}(\mathfrak P)\le\mathrm{non}(\mathcal N)$?

Here $\mathrm{non}(\mathcal N)$ is the smallest cardinality of a subset of the real line, which is not Lebesgue null.