Maximal free sequences in a Boolean algebra

Monk, J. D.

Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 593-610.

Résumé

We study free sequences and related notions on Boolean algebras. A free sequence on a BA $A$ is a sequence $\langle a_\xi:\xi \alpha \rangle$ of elements of $A$, with $\alpha$ an ordinal, such that for all $F,G\in[\alpha]^{\omega}$ with $F$ we have $\prod_{\xi\in F}a_\xi\cdot \prod_{\xi\in G}-a_\xi \not=0$. A free sequence of length $\alpha$ exists iff the Stone space $\operatorname{Ult}(A)$ has a free sequence of length $\alpha $ in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function $$ {\frak f}_{\operatorname{sp}}(A)=\{|\alpha|:A\hbox{ has an infinite maximal free sequence of length }\alpha \} $$ and the associated min-max function $$ {\frak f}(A)=\min({\frak f}_{\operatorname{sp}}(A)). $$ Among the results are: for infinite cardinals $\kappa\leq\lambda$ there is a BA $A$ such that ${\frak f}_{\operatorname{sp}}(A)$ is the collection of all cardinals $\mu$ with $\kappa\leq\mu\leq\lambda$; maximal free sequences in $A$ give rise to towers in homomorphic images of $A$; a characterization of ${\frak f}_{\operatorname{sp}}(A)$ for $A$ a weak product of free BAs; ${\frak p}(A), \pi\chi_{\inf}(A)\leq{\frak f}(A)$ for $A$ atomless; a characterization of infinite BAs whose Stone spaces have an infinite maximal free sequence; a generalization of free sequences to free chains over any linearly ordered set, and the relationship of this generalization to the supremum of lengths of homomorphic images.

Détail

Classification : 06E05, 06E15, 54A25
Keywords: free sequences; cardinal functions; Boolean algebras