2
$\begingroup$

For $A,B\in {\cal P}(\omega)$ let us say that $A\simeq_{\rm{fin}} B$ if both $A\setminus B$ and $B\setminus A$ are finite. It is easy to see that this establishes an equivalence relation on ${\cal P}(\omega)$. Moreover, we say $A\subseteq^* B$ if $A\setminus B$ is finite. This relation can be used to impose an ordering on the set of equivalence classes on ${\cal P}(\omega)$; we denote that ordered set by ${\cal P}(\omega)/\rm{fin}$.

Question. If $P, Q$ are partially ordered sets with $P\times Q \cong {\cal P}(\omega)/\rm{fin}$, does this imply that one of $P, Q$ consists of $1$ element only?

Improve this question
$\endgroup$
6
  • 5
    $\begingroup$ ${\cal P}(\omega)/\rm{fin}\times{\cal P}(\omega)/\rm{fin}\cong{\cal P}(\omega)/\rm{fin}$. $\endgroup$
    – Wojowu
    Commented Oct 17, 2022 at 12:13
  • $\begingroup$ @Wojowu: You sure about that? $\endgroup$
    – Will Brian
    Commented Oct 17, 2022 at 12:15
  • $\begingroup$ @WillBrian If I'm not mistaken, we have an isomorphism by taking an "interlacing" map - given a pair of equivalence classes $[A],[B]$ we map it to the class $[2A\cup (2B+1)]$ (where $2A$ and $2B+1$ are defined element-wise). It seems to me to be an isomorphism of posets as well. $\endgroup$
    – Wojowu
    Commented Oct 17, 2022 at 12:21
  • 4
    $\begingroup$ The Boolean algebra $A=\mathcal{P}(\omega)/\mathrm{fin}$ is isomorphic to $A^2$. This reflects the fact that the Stone-Cech remainder $X$ of $\omega$ is homeomorphic to the disjoint union $X\sqcup X$ (this is quite obvious from either point of view). $\endgroup$
    – YCor
    Commented Oct 17, 2022 at 13:03
  • 3
    $\begingroup$ If $B$ is a Boolean algebra, and $p$ is a finite partition of $B$ (by partition, I mean that $\bigvee p=1$ and $a\wedge b=0$ for $a,b\in p,a\neq b$), then $B\simeq\prod_{a\in p}B\upharpoonright a$ where $B\upharpoonright a=\{b\in B\mid b\leq a\}.$ This does not look like research level mathematics. Or am I missing something? $\endgroup$
    – Joseph Van Name
    Commented Oct 17, 2022 at 13:20

0

Reset to default

Browse other questions tagged .