%Version of April 10, 2001

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\title{On the cofinalities of Boolean algebras and the ideal of null sets}

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\markboth{K.~Ciesielski and J.~Pawlikowski
%\ \ \ \ \ \today
}{On the cofinalities of Boolean algebras and the ideal of null sets
%\ \ \ \ \ \today
}

 
\author{
Krzysztof Ciesielski%
\thanks{%\endgraf 
%%%The work of the first author was partially supported by 
%%%NSF Cooperative
%%%Research Grant INT-9600548, with its Polish part %being 
%%%financed by Polish Academy of Science PAN.\endgraf
AMS classification numbers: Primary 06E25;
Secondary  03E17, 03G05. \endgraf
Key words and phrases: cofinality, null sets.}
\\
{\footnotesize Department of Mathematics,}
{\footnotesize West Virginia University,} \\
{\footnotesize Morgantown, WV 26506-6310, USA}\\
{\footnotesize e-mail: K\_Cies@math.wvu.edu}; %\\
{\footnotesize web page: {\tt http://www.math.wvu.edu/\~{}kcies}}
\and
Janusz Pawlikowski\\
{\footnotesize Department of Mathematics,}
{\footnotesize University of  Wroc\l aw,} \\
{\footnotesize pl. Grunwaldzki 2/4, 50-384 Wroc\l aw, Poland;} %\\
{\footnotesize e-mail: pawlikow@math.uni.wroc.pl}\\
{\footnotesize and}\\
{\footnotesize Department of Mathematics,}
{\footnotesize West Virginia University,}\\
{\footnotesize Morgantown, WV 26506-6310, USA; }%\\
{\footnotesize e-mail: pawlikow@math.wvu.edu}
}

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\begin{document}
 
\maketitle



\begin{abstract}
We will show that if the cofinality of the ideal of 
Lebesgue measure zero sets is equal to $\omega_1$ then 
there exists a Boolean algebra $B$ of cardinality
$\omega_1$ which is not
a union of strictly increasing $\omega$-sequence
of its subalgebras. This generalizes a result of Just and Koszmider
who showed that it is consistent with 
ZFC+$\neg$CH that such an algebra exists.
\end{abstract}

%\newpage 


\section{Preliminaries}

For an infinite Boolean algebra $B$ 
its {\it cofinality}\/ $\cf(B)$ is defined 
as the least infinite cardinal number $\kappa$ such that
$B$ is a union of strictly increasing sequence of type $\kappa$ 
of subalgebras of $B$; its {\it homomorphism type}\/ $h(B)$ is 
the least cardinality of an infinite homomorphic image of $B$.  

In~\cite{Kopp} Koppelberg proved that 
\begin{itemize}
\item[(a)] $\omega\leq\cf(B)\leq h(B)\leq\continuum$, and 
\item[(b)] if Martin's Axiom holds then $\cf(B)=\omega$ for every
Boolean algebra $B$ with $|B|<\continuum$; in particular 
$h(B)\in\{\omega,\continuum\}$ for every Boolean algebra $B$.
\end{itemize}
(See also \cite{JuKosz} and \cite{vDMR}.)

In~\cite{JuKosz} Just and Koszmider
examined a question whether in (b) the assumption 
of Martin's Axiom is important. They gave a positive answer to it by 
showing that there exists a model of ZFC 
(obtained by adding Sacks reals side-by-side)
in which there is a Boolean algebra  $B$  such that
$|B|=\cf(B)=\omega_1<\continuum$. 
Clearly for this algebra we have also $h(B)=\omega_1\notin\{\omega,\continuum\}$
since $\cf(B)\leq h(B)\leq |B|$. 

The goal of this paper is to prove the following theorem, in which
$\NN$ denotes the ideal of Lebesgue measure subset of
$\real$ and $\cof(\NN)$ its cofinality, that is, 
\[
\cof(\NN)=\min\{|\B|\colon \B\subset\NN\mbox{ generates } \NN\}.
\]


\thm{thJustKosz}{$\cof(\NN)=\omega_1$ implies that 
there exists a Boolean algebra $B$ of cardinality
$\omega_1$ such that $\cof(B)=\omega_1$.
}

This generalizes the result of Just and Koszmider
since in the model they worked with $\cof(\NN)=\omega_1$ holds. 
However, $\cof(\NN)=\omega_1$ holds in many other models as well.
(For example, it follows from the Covering Property Axiom CPA
as shown by the authors in~\cite{CPAbook}.)
Moreover, the argument presented here is considerably 
simpler than the one from~\cite{JuKosz}.

Our set theoretic terminology is standard and follows that of~\cite{CiBook}.
In particular, $|X|$ stands for the cardinality of a set $X$
and $\continuum=|\real|$. 
In what follows we will use the following characterization of $\cf(\NN)$,
in which $\C_H$ 
stands for the family of all subsets $\prod_{n<\omega}T_n$
of $\omega^\omega$ such that $T_n\in[\omega]^{\leq n+1}$ for all $n<\omega$.

\prop{propBarto}{{\rm (Bartoszy\'nski~\cite[thm. 2.3.9]{BJ})}
\[
\cof(\NN)=
\min\left\{|\F|\colon \F\subset\C_H\ \&\ \bigcup\F=\omega^\omega\right\}.
\]
}



\section{The proof}



The proof of Theorem~\ref{thJustKosz} will be based on the following lemma.

\lem{lemJustKosz}{If $\cof(\NN)=\omega_1$ then for every infinite 
countable Boolean algebra $\A$ there exists 
a family $\{a^\xi_n\in\A\colon n<\omega\ \&\ \xi<\omega_1\}$
with the following properties.
\begin{itemize}
\item[{\rm (i)}] $a^\xi_n\wedge a^\xi_m={\bf 0}$ for every 
           $n<m<\omega$ and $\xi<\omega_1$.
\item[{\rm (ii)}] For every increasing sequence
$\la\A_n\colon n<\omega\ra$ of proper subalgebras of $\A$ with 
$\A=\bigcup_{n<\omega}\A_n$ there exists a $\xi<\omega_1$
such that $a^\xi_n\notin \A_n$ for all $n<\omega$. 
\end{itemize}
}
\proof In the argument that follows every sequence 
$\bar\A=\la\A_n\colon n<\omega\ra$
as in (ii) will be identified with a function
$f_{\bar\A}=f\in\omega^\A$ for which 
$f^{-1}(n)=\A_n\setminus\bigcup_{i<n}\A_i$. 
We will denote the set of all such functions by $X$. 
Also, let $\{b_n\colon n<\omega\}$ be an enumeration 
of $\A$ and for each $n<\omega$ let $B_n$ be a finite algebra 
generated by $\{b_i\colon i<n\}$. 
Thus, $\A=\bigcup_{i<n}B_i$.

Since $\cof(\NN)=\omega_1$, the dominating number 
\[
{\mathfrak d}=\min\left\{|\K|\colon \K\subset\omega^\omega\ \&\ 
      (\forall f\in \omega^\omega) (\exists g\in\K)
      (\forall n<\omega)\ f(n)<g(n)\right\}
\]
is equal to $\omega_1$. (See e.g.~\cite{BJ}.)
So, 
there exists a dominating family $\K\subset\omega^\omega$
of cardinality $\omega_1$. We can also assume that 
the sequences in $\K$ are strictly increasing
and that for every $g\in\K$ function $\bar g$ defined by
$\bar g(n)=\sum_{i\leq n}g(i)$ also belongs to $\K$. 

Next note that for every $f\in X$ there are 
$\bar d=\la d_k\colon k<\omega\ra\in\K$
and $\bar r=\la r_k\colon k<\omega\ra\in\K$
such that for every $k<\omega$
\begin{itemize}
\item[(a)] $f(b)< r_k$ for all $b\in B_k$; and 

\item[(b)] there are %exist 
disjoint $b_0,\ldots,b_{2k}\in B_{d_k}$ with %such that
   $r_{d_{k-1}}<f(b_0)<\cdots<f(b_{2k})$.
\end{itemize}
Indeed, the existence of $\bar r$ satisfying (a) 
follows directly from the definition of a dominating family.
Moreover, 
since all algebras $\A_n=f^{-1}(\{0,\ldots,n\})$ are proper,
for every number $d<\omega$ there exist
disjoint $b_0,\ldots,b_{2d}\in\A$ such that
$r_d<f(b_0)<\cdots<f(b_{2d})$.
Let $h\in\omega^\omega$ be such that 
$b_0,\ldots,b_{2d}\in B_{d+h(d)}$ for every $d<\omega$
and let $g\in\K$ be a function dominating $h$.
Then $\bar d=\bar g$ is as required. 

The above implies, in particular, that 
for every $f\in X$ there are $\bar d,\bar r\in\K$
such that $f$ satisfies (b) and the sequence 
$f^{\bar r}=\la f\restriction (B_{d_k}\setminus B_{d_{k-1}})\colon k<\omega\ra$
belongs to 
\[
X({\bar d},{\bar r})=
\prod_{k<\omega}(r_{d_k})^{B_{d_k}\setminus B_{d_{k-1}}}.
\]

Now, since $\cof(\NN)=\omega_1$, by Proposition~\ref{propBarto}
(applied to $\prod_{k<\omega}\omega^{B_{d_k}\setminus B_{d_{k-1}}}$
in place of $\omega^\omega$) we can find 
an $\omega_1$-covering of $X({\bar d},{\bar r})$
by sets $T$ of the form 
$\prod_{k<\omega}T_k$, where 
$T_k\in\left[\omega^{B_{d_k}\setminus B_{d_{k-1}}}\right]^{\leq k+1}$ 
for all $k<\omega$.
Since the total number of these sets $T$ (for different $\bar d,\bar r\in\K$)
is equal to $\omega_1$, 
to finish the proof it is enough to show
that for any such $T$ there is one sequence
$\la a_n\colon n<\omega\ra$ satisfying (i) and such that (ii)
holds for every 
for every 
$\bar\A=\la\A_n\colon n<\omega\ra$
for which $f^{\bar r}_{\bar\A}$ belongs to $T$ and 
$f_{\bar\A}$ satisfies (b). 

So, let $T$ be as above and let $T^*$ be the set of all 
functions $f_{\bar\A}$
satisfying (b) for which $f^{\bar r}_{\bar\A}\in T$.
By induction on $k<\omega$ we will construct 
a sequence 
$\la c_k\in{B_{d_k}\setminus B_{d_{k-1}}}\colon k<\omega\ra$
such that 
\begin{itemize}
\item [($*$)] 
$f(c_k)> r_{d_k}\geq k$ for every $k<\omega$ and $f\in T^*$. 
\end{itemize}

So fix a $k<\omega$ and let $\{f_i\colon i<k\}$
be such that
\[
\{f_i\restriction{B_{d_k}\setminus B_{d_{k-1}}}\colon i<k\}=
\{f\restriction{B_{d_k}\setminus B_{d_{k-1}}}\colon f\in T^*\}\subset T_k.
\]
We show inductively that for every $m< k$ 
\begin{equation}\label{eqJK}
\mbox{there is 
a $c\in B_{d_k}$ such that $f_j(c)>r_{d_k}$ for all $j\leq m$.}
\end{equation}

So, fix an $m< k$ and let 
$a\in B_{d_k}$ such that $f_j(a^c)=f_j(a)>r_{d_k}$ for all $j< m$. 
If $f_m(a)>r_{d_k}$ then $c=a$ satisfies (\ref{eqJK}). 
Thus, assume that $f_m(a^c)=f_m(a)\leq r_{d_k}$. 
By (b) we can find $b_0,\ldots,b_{2k}\in B_{d_k}$ such that
$r_{d_{k-1}}<f_m(b_0)<\cdots<f_m(b_{2k})$.
By Pigeon Hole Principle we can find an $I\in[\{0,\ldots,2k\}]^{k+1}$ 
and a $b\in\{a,a^c\}$ such that 
$f_m(b\wedge b_i)=f_m(b_i)$ for all $i\in I$.
Without loss of generality we can assume that 
$I=\{0,\ldots,k\}$ and $b\wedge b_i=b_i$ for all $i\leq k$.
Then
\[
f_m(b^c\vee b_i)>r_{d_k}\ \mbox{ for all } i\leq k.
\]
Moreover, for every $j<m$ there is at most one $i_j\leq k$ for which
\[
f_j(b^c\vee b_{i_j})\leq r_{d_k}
\]
since for different $i,i'\leq k$ we have
$f_j((b^c\vee b_i)\wedge(b^c\vee b_{i'}))=f_j(b^c)>r_{d_k}$. 
Thus, by Pigeon Hole Principle, there is an $i\leq k$
such that $c=b^c\vee b_i$ satisfies (\ref{eqJK}).
This finishes the proof of ($*$). 

Clearly the sequence 
$\la c_k\in{B_{d_k}\setminus B_{d_{k-1}}}\colon k<\omega\ra$
satisfies (ii) for every $\bar\A$ with 
$f_{\bar\A}\in T^*$. Thus, we need only to modify 
it to get also the condition (i). 

To do it, use the fact that
\begin{center}
$r_{d_k}<f(c_k)<r_{d_{k+1}}$ for every $k<\omega$ and $f\in T^*$
\end{center}
to construct the sequences:
$\omega=I_0\supset I_1\supset \cdots$ of infinite subsets of $\omega$,
increasing $\la k_j\in I_j\colon j<\omega\ra$, and 
$\la c_{k_j}^*\in\{c_{k_j},c_{k_j}^c\}\colon j<\omega\ra$ such that
for every $j<\omega$ 
\begin{itemize}
\item $f(\bar a_{k_j}\wedge c_l)>r_{d_l}$ for every $f\in T^*$
       and $l>k_j$ with $l\in I_j$, 
\end{itemize}
where $\bar a_{k_j}=c_{k_0}^*\wedge \cdots \wedge c_{k_j}^*$. 
Then the sequence $\la \bar a_{k_j}\colon j<\omega\ra$
is a strictly decreasing sequence
satisfying (ii) and it is now easy to see that by 
putting $a_j=\bar a_{k_j}\wedge \bar a_{k_{j+1}}^c$
we obtain the desired sequence. \qed

\medskip 

\noindent{\sc Proof of Theorem~\ref{thJustKosz}.} 
Algebra $B$ we construct will be a subalgebra
of the algebra $\P(\omega)$ of all subsets of $\omega$. 
First, let $\K\subset\omega^\omega$ be a dominating family
with $|\K|=\omega_1$ and fix a partition 
$\{D_k\colon k<\omega\}$ of $\omega$ into infinite subsets. 

For every sequence $\bar a=\la a_n\colon n<\omega\ra$ 
of pairwise disjoint subsets of $\omega$ and $k<\omega$
put
$a_k^*=\bigcup\{a_n\colon n\in D_k\}$. 
In addition, for every $h\in\K$ 
we put 
\[
a^h=\bigcup\{a_{n_h(k)}\colon k<\omega\}
\]
where $n_h(k)=\min\{n\in D_k\colon n>\max\{h(k),k\}\}$.
We also put 
\[
F(\bar a)=\{a_k^*\colon k<\omega\}\cup\{a^h\colon h\in\K\}
\in[\P(\omega)]^{\leq\omega_1}.
\]

Next, we will construct an increasing sequence
$\la B_\xi\in[\P(\omega)]^{\omega_1}
\colon \xi\leq\omega_1\ra$ of subalgebras
of $\P(\omega)$ aiming for $B=B_{\omega_1}$.
Thus, we choose $B_0$ as an arbitrary 
subalgebra of $\P(\omega)$ with $|B_0|=\omega_1$
and for limit ordinal numbers $\lambda\leq\omega_1$
we put $B_\lambda=\bigcup_{\xi<\lambda}B_\xi$.
The algebra $B_{\xi+1}$ is formed from 
$B_\xi$ in the following way.

Let $\{b_\eta\colon\eta<\omega_1\}$ be an enumeration
of $B_\xi$ and for $\eta<\omega$ let 
$\A^\xi_\eta$ be a subalgebra of $B_\xi$ 
generated by $\{b_\zeta\colon\zeta<\eta\}$.
For each such algebra we apply 
Lemma~\ref{lemJustKosz} to find the sequences 
$\bar a^\gamma=\la a^\gamma_n\colon n<\omega\ra$,
$\gamma<\omega_1$, satisfying (i) and (ii)
and let
\[
G(\A^\xi_\eta)=\bigcup_{\gamma<\omega_1} F(\bar a^\gamma).
\]
$B_{\xi+1}$ is defined as the algebra generated by
$B_\xi\cup\bigcup_{\eta<\omega_1}G(\A^\xi_\eta)$. 
This finishes the construction of $B$. 

Clearly, $|B|=\omega_1$. To prove that $\cof(B)=\omega_1$
it is enough to show that $B$ is not a union of an increasing sequence 
$\bar\B=\la B_n\colon n<\omega\ra$ of proper subalgebras. 
So, by way of contradiction,
assume that such a sequence $\bar\B$ exists.
For every $n<\omega$ choose 
$b_n\in B\setminus B_n$ and find $\xi,\eta<\omega_1$ such that
$\{b_n\colon n<\omega\}\subset \A^\xi_\eta$. 
Then the algebras $\A_n=B_n\cap \A^\xi_\eta$
form an increasing sequence of proper subalgebras of $\A=\A^\xi_\eta$.
Thus, one of the sequences 
$\bar a^\gamma$
satisfies (ii) for $\bar\A$.
So, if we put $\bar a^\gamma=\bar a=\la a_n\colon n<\omega\ra$
we conclude that
$\{a_k^*\colon k<\omega\}\cup\{a^h\colon h\in\K\}\subset B$.
Let $f(k)=\min\{n<\omega\colon a_k^*\in B_n\}$ and let 
$h\in\K$ be such that $f(k)<h(k)$ for all $k<\omega$. 

The final contradiction is obtained by noticing that 
$a^h$ cannot belong to any $B_k$.
Indeed, if $a^h\in B_k$ for some $k$, then 
$a^h\cap a^*_k=a_{n_h(k)}$
belongs to $B_{\max\{f(k),k\}}$, since $a^*_k\in B_{f(k)}$.
But $\max\{f(k),k\}\leq\max\{h(k),k\}<n_h(k)$ so we get
$a_{n_h(k)}\in B_{n_h(k)}$
contradicting the fact that $a_{n_h(k)}$
belongs to $\bar\A\setminus \A_{n_h(k)}$,
which is disjoint with $B_{n_h(k)}$.
\qed






\begin{thebibliography}{22}
\addcontentsline{toc}{chapter}{Bibliography}

\bibitem{BJ} T.~Bartoszy\'{n}ski, H.~Judah, 
{\em Set Theory}, A~K~Peters, 1995.

\bibitem{CiBook} K.~Ciesielski,
{\it Set Theory for the Working Mathematician}, 
London Math. Soc. Stud. Texts {\bf 39}, Cambridge Univ. Press 1997.


\bibitem{CPAbook} K.~Ciesielski, J.~Pawlikowski,
{\it Covering Property Axiom CPA}, 
version of March 2001, work in progress
available form the 
{\it Set Theoretic Analysis Web Page:}
http://www.math.wvu.edu/\~{}kcies/STA/STA.html

%preprint$^\star$.%
%\footnote{Preprints marked by $^\star$ are available in electronic form
%from {\it Set Theoretic Analysis Web Page:}
%http://www.math.wvu.edu/\~{}kcies/STA/STA.html}


\bibitem{vDMR} E.~K.~van Douwen, J.~D.~Monk, M.~Rubin,
{\it Some questions about Boolean algebras},
Algebra Universalis {\bf 11} (1980), 220--243.


\bibitem{JuKosz} W.~Just, P.~Koszmider, 
{\it Remarks on cofinalities and homomorphism types of Boolean algebras},
Algebra Universalis {\bf 28}(1) (1991), 138--149.

\bibitem{Kopp} S. Koppelberg, 
{\it Boolean algebras as unions of chains of subalgebras},
Algebra Universalis {\bf 7} (1977), 195--204.




\end{thebibliography}

\end{document}

