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Results tagged with boolean-algebras
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user 5903
A Boolean algebra is a commutative ring satisfying x²=x for every x, and sometimes required to have a unit; they have characteristic 2. For coding theory (notably dealing with subsets linear subspaces of spaces of Boolean functions), rather use the [coding-theory] or [linear-algebra] tag.
4
votes
Is there such a thing as the sigma-completion of a Boolean algebra?
This question was answered in topological terms by J. Vermeer in The smallest basically disconnected preimage of a space. Topology Appl. 17 (1984), no. 3, 217–232.
See here for a review and here for t …
- 11.4k
3
votes
Chain of ideals in a BA
Take te interval algebra, $B$, of the ordinal $\omega_1\times\omega$; for $n\in\omega$ let $I_n$ be the ideal consisting of the elements of $B$ that are bounded below $\omega_1\times n$.
None of the i …
- 11.4k
2
votes
Can this ultrafilter convergence condition be expressed as a compactness condition?
Your condition amounts to saying that the uniformity $\mathcal{C}$ generated by all clopen partitions is complete: 'partition-prime' is the same as '$\mathcal{C}$-Cauchy'.
It is also implied by the co …
- 11.4k
16
votes
Accepted
Is Stone-Čech compactification of 0-dimensional space also 0-dimensional?
In this paper (Spaces $N\cup\mathscr{R}$ and their dimensions, Topol. Appl.
11(1) 1980 93-102) — I hope the PDF is freely available) Jun Terasawa constructs maximal almost disjoint families on $\mathb …
- 11.4k
2
votes
Accepted
Description of atomless complete Boolean algebras with a countable $\pi$-base
To make the answers concrete: the Boolean algebra of clopen subsets of the Cantor set is the unique countable atomless Boolean algebra.
Its completion is the regular-open algebra of the Cantor set, wh …
- 11.4k
2
votes
Accepted
Complete CCC Boolean algebras (or Stonean spaces)
A partial answer to the question about the Gleason cover of the unit interval: it can be found everywhere in every compact extremally disconnected space.
The point is: in a compact extremally disconne …
- 11.4k
9
votes
Accepted
A strictly descending chain of subalgebras of $P(\omega)/_{\mathrm{fin}}$
There is a family $\{K_X:X\subseteq\mathfrak{c}\}$ of separable compact
zero-dimensional spaces such that there is a continuous surjection of $K_X$
onto $K_Y$ if and only if $X\subseteq Y$.
These spac …
- 11.4k
1
vote
About the existence of a particular kind of "splitting" function on atomless complete Boolea...
I think you are asking too much.
Assume we have such a function and let $a$ be nonzero such that both $a_0$ and $a_1$ are nonzero. Then $b\le a_0$ implies $b_1=0$ and $b\le a_1$ implies $b_0=0$.
If …
- 11.4k
4
votes
Accepted
Extremally disconnected rigid infinite Hausdorff compacta(?)
This paper, Rigid Stone spaces within $\mathsf{ZFC}$ by Dow, Gubbi, and Szymański contains relatively easy examples of such spaces.
- 11.4k
6
votes
A unique ultrafilter extending a union of filters?
The following is due to Alan Dow:
In any model obtained by adding $\aleph_2$ many Cohen reals to a model
of $\mathsf{CH}$ the statement is false.
We force with $\mathbb{P}=\operatorname{Fn}(\omega_2, …
- 11.4k
3
votes
Ideals on $\mathbb N$ and large sets that have small intersection
In topological language: for any closed subset, $F$, of $\beta\mathbb{N}\setminus\mathbb{N}$ the family $I_F=\{A:A^*\cap F=\emptyset\}$ is an ideal.
Your property translates into: the closed set $F$ i …
- 11.4k
5
votes
Self-homeomorphism of Stone-Čech boundary with an isolated fixed point
As an answer to the bonus question: no, see K. P. Hart and J. Vermeer. Fixed-point sets of autohomeomorphisms of compact F-spaces,
Proceedings of the American Mathematical Society, 123 (1995), 311–314 …
- 11.4k
5
votes
Accepted
A problem of non-emptiness of intersections of certain chains of regular open sets
Here is a provisional negative answer.
If $\mathcal{C}$ is a c-minimal chain then $N=\bigcap\mathcal{C}$ is closed and nowhere dense (if nonempty): if the interior is nonempty take a nonempty regular …
- 11.4k
6
votes
Accepted
Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?
No, $\mathrm{RO}(X)$ is complete; $\mathcal{P}(\omega)/\mathit{fin}$ is not (no strictly increasing sequence has a supremum).
- 11.4k
4
votes
Accepted
Copy of $P(\omega)/\mathrm{fin}$ on $\omega_1$
To expand my comment into an answer: take, for each $n\in\omega$, a uniform ultrafilter $u_n$ on $\omega_1$ that contains the set $\{\lambda+n:\lambda$ is a limit or $0\}$.
The set $U=\{u_n:n\in\omega …
- 11.4k