%Version of April 10, 2001 \documentclass[12pt]{article} \usepackage{amssymb} \date{} \title{On the cofinalities of Boolean algebras and the ideal of null sets} %\pagestyle{headings} \pagestyle{myheadings} \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} } \newcommand{\new}{\marginpar{{\tiny NEW}}} \newcommand{\ch}[1]{\marginpar{{\tiny #1}}} \newcounter{ChartNo} \newcommand{\Ccounter}{\refstepcounter{ChartNo}\theChartNo} \newcommand{\forces}{\mathrel{\|}\joinrel\mathrel{-}} \newcommand{\dec}{{\rm dec}} \newcommand{\ignore}[1]{} \newcommand{\IntTh}{{\rm IntTh}} \newcommand{\Implies}{\Longrightarrow} \newcommand{\SoIC}{{s_0^{\rm prism}}} \newcommand{\SoST}{{s_0^{\rm cube}}} \newcommand{\psm}{{\rm CPA}} \newcommand{\psmC}{\mbox{{\rm CPA$_{\rm cube}$}}} \newcommand{\psmP}{\mbox{{\rm CPA$_{\rm prism}$}}} \newcommand{\psmCsec}{\mbox{{\rm CPA$_{\rm cube}^{\rm sec}$}}} \newcommand{\psmPsec}{\mbox{{\rm CPA$_{\rm prism}^{\rm sec}$}}} 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\newcommand{\fact}[2]{\begin{Fact}\label{#1}{\sl #2}\end{Fact}} \newcommand{\claim}[2]{\begin{Claim}\label{#1}{\sl #2}\end{Claim}} \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 $n r_{d_k}\geq k$ for every $k<\omega$ and $f\in T^*$. \end{itemize} So fix a $k<\omega$ and let $\{f_i\colon ir_{d_k}$ 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}}r_{d_k}\ \mbox{ for all } i\leq k. \] Moreover, for every $jr_{d_k}$. 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} $r_{d_k}r_{d_l}$ 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)