Maximal free sequences in a Boolean algebra

Monk, J. D.. "Maximal free sequences in a Boolean algebra." Commentationes Mathematicae Universitatis Carolinae 52.4 (2011): 593-610. <http://eudml.org/doc/247226>.

@article{Monk2011,
abstract = {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 $F<G$ we have $\prod _\{\xi \in F\}a_\xi \cdot \prod _\{\xi \in G\}-a_\xi \ne 0$. 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 \[ \{\mathfrak \{f\}\}\_\{\operatorname\{sp\}\}(A)=\lbrace |\alpha |:A\hbox\{ has an infinite maximal free sequence of length \}\alpha \rbrace \] and the associated min-max function \[ \{\mathfrak \{f\}\}(A)=\min (\{\mathfrak \{f\}\}\_\{\operatorname\{sp\}\}(A)). \] Among the results are: for infinite cardinals $\kappa \le \lambda $ there is a BA $A$ such that $\{\mathfrak \{f\}\}_\{\operatorname\{sp\}\}(A)$ is the collection of all cardinals $\mu $ with $\kappa \le \mu \le \lambda $; maximal free sequences in $A$ give rise to towers in homomorphic images of $A$; a characterization of $\{\mathfrak \{f\}\}_\{\operatorname\{sp\}\}(A)$ for $A$ a weak product of free BAs; $\{\mathfrak \{p\}\}(A), \pi \chi _\{\inf \}(A)\le \{\mathfrak \{f\}\}(A)$ 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.},
author = {Monk, J. D.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {free sequences; cardinal functions; Boolean algebras; Boolean algebra; cardinal function; free sequence},
language = {eng},
number = {4},
pages = {593-610},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Maximal free sequences in a Boolean algebra},
url = {http://eudml.org/doc/247226},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Monk, J. D.
TI - Maximal free sequences in a Boolean algebra
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 4
SP - 593
EP - 610
AB - 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 $F<G$ we have $\prod _{\xi \in F}a_\xi \cdot \prod _{\xi \in G}-a_\xi \ne 0$. 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 \[ {\mathfrak {f}}_{\operatorname{sp}}(A)=\lbrace |\alpha |:A\hbox{ has an infinite maximal free sequence of length }\alpha \rbrace \] and the associated min-max function \[ {\mathfrak {f}}(A)=\min ({\mathfrak {f}}_{\operatorname{sp}}(A)). \] Among the results are: for infinite cardinals $\kappa \le \lambda $ there is a BA $A$ such that ${\mathfrak {f}}_{\operatorname{sp}}(A)$ is the collection of all cardinals $\mu $ with $\kappa \le \mu \le \lambda $; maximal free sequences in $A$ give rise to towers in homomorphic images of $A$; a characterization of ${\mathfrak {f}}_{\operatorname{sp}}(A)$ for $A$ a weak product of free BAs; ${\mathfrak {p}}(A), \pi \chi _{\inf }(A)\le {\mathfrak {f}}(A)$ 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.
LA - eng
KW - free sequences; cardinal functions; Boolean algebras; Boolean algebra; cardinal function; free sequence
UR - http://eudml.org/doc/247226
ER -