| Title:
|
A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube (English) |
| Author:
|
Kurilić, Miloš S. |
| Author:
|
Pavlović, Aleksandar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
519-537 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We compare the forcing-related properties of a complete Boolean algebra ${\mathbb B}$ with the properties of the convergences $\lambda _{\mathrm s}$ (the algebraic convergence) and $\lambda _{\mathrm {ls}}$ on ${\mathbb B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that $\lambda _{\mathrm {ls}}$ is a topological convergence iff forcing by ${\mathbb B}$ does not produce new reals and that $\lambda _{\mathrm {ls}}$ is weakly topological if ${\mathbb B}$ satisfies condition $(\hbar )$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda _{\mathrm {ls}}$ is a weakly topological convergence, then ${\mathbb B}$ is a $2^{\mathfrak h}$-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence $\lambda _{\mathrm {ls}}$ on the collapsing algebra ${\mathbb B}=\mathop {\mathrm {ro}} (^{<\omega }\omega _2)$ is weakly topological“ is independent of ZFC. (English) |
| Keyword:
|
complete Boolean algebra |
| Keyword:
|
convergence structure |
| Keyword:
|
algebraic convergence |
| Keyword:
|
forcing |
| Keyword:
|
Cantor cube |
| Keyword:
|
Aleksandrov cube |
| Keyword:
|
small cardinal |
| MSC:
|
03E17 |
| MSC:
|
03E40 |
| MSC:
|
06E10 |
| MSC:
|
54A20 |
| MSC:
|
54D55 |
| idZBL:
|
Zbl 06391510 |
| idMR:
|
MR3277752 |
| DOI:
|
10.1007/s10587-014-0117-6 |
| . |
| Date available:
|
2014-11-10T09:57:35Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144014 |
| . |
| Reference:
|
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