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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 $\mathfrak P$.

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  • $\begingroup$ Re: Compatibility: Two partitions ${\cal A}, {\cal B}$ are compatible if there is a directed family ${\frak D}$ of partitions with respect to the refinement ordering and ${\cal A}, {\cal B}\in{\frak D}$ - is that correct? (Just trying to make a link to a previous question of yours that involved directed families of partition orderings.) $\endgroup$ Feb 18 '20 at 8:06
  • $\begingroup$ Yes, exactly! This is equivalent. $\endgroup$ Feb 18 '20 at 9:37
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    $\begingroup$ Doesn't "cardinal-characteristics" make "independence-results" redundant? while it's a pity not to tag it "infinite-combinatorics" where this question primarily belongs. $\endgroup$
    – YCor
    Feb 20 '20 at 7:46
  • $\begingroup$ 23rd version of the question. $\endgroup$ Feb 26 '20 at 5:57
  • $\begingroup$ @GerryMyerson Simply I am thinking on this problem myself, so some information appears, which I add to the questions in edits. For example, Problem 3 disappeared as I was able to show that ${\uparrow}{\downarrow}(\mathfrak P)=\mathfrak j_2=\mathfrak j_{2:1}$ thus reducing the problem to another my question about $\mathfrak j_{2:1}$. Since there was no answers, I hope such revisions of the problems are not grave. But anyway, thank you for turning attention to this my problem. $\endgroup$ Feb 26 '20 at 6:07
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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.

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  • $\begingroup$ I’m confused. What is $\downarrow\downarrow$ and $\uparrow\uparrow$? And how does Theorem 6.1 give ${\downarrow}(\mathfrak P)=\mathfrak c$? $\endgroup$ Mar 4 '20 at 9:57
  • $\begingroup$ @EmilJeřábek3.0 Ups! Thank you for the comment. I have just copy-pasted the theorem from my preprint and there were two more cardinal characteristics ${\uparrow\uparrow}(\mathfrak P)$ and ${\downarrow\downarrow}(\mathfrak P)$, which are analogs of the cellularity. Unfortunately, arXiv is a bit slow and the new version of the paper with all these symbols will appear only tomorrow (I hope). Till that moment the new version of the paper with proper form of Theorem 6.1 can be found at researchgate.net/publication/… $\endgroup$ Mar 4 '20 at 11:12
  • $\begingroup$ Great, thank you. $\endgroup$ Mar 4 '20 at 11:25

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