| Title:
|
Maximal free sequences in a Boolean algebra (English) |
| Author:
|
Monk, J. D. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
593-610 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\prod_{\xi\in F}a_\xi\cdot \prod_{\xi\in G}-a_\xi \not=0$. 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. (English) |
| Keyword:
|
free sequences |
| Keyword:
|
cardinal functions |
| Keyword:
|
Boolean algebras |
| MSC:
|
06E05 |
| MSC:
|
06E15 |
| MSC:
|
54A25 |
| idZBL:
|
Zbl 1249.06034 |
| idMR:
|
MR2864001 |
| . |
| Date available:
|
2011-12-16T10:11:17Z |
| Last updated:
|
2015-02-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141738 |
| . |
| Reference:
|
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| Reference:
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