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Results tagged with boolean-algebras
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user 7206
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.
11
votes
Linear suborders of $(P(\omega),\subseteq)$
Note that the lexicographic order of $2^\omega$ is a linear extension of $(\mathcal P(\omega),\subseteq)$, namely if $A\subseteq B$, then $\min(A\mathbin{\triangle}B)\in B$, which means that the first …
- 39.9k
8
votes
1
answer
560
views
On $V$-decisive and weakly homogeneous forcings
Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb …
- 39.9k
7
votes
3
answers
437
views
Chain conditions in quotients of power sets
Several days ago a friend asked me the following:
We know that in $\mathcal P(\mathbb N)$ we can find a family of size continuum that every [distinct] two intersect in a finite set. Can we do that …
- 39.9k
5
votes
2
answers
396
views
How complete is $\mathcal P(\kappa)/J_{bd}$?
While it is true that $\mathcal P(\kappa)$ is a complete Boolean algebra, it is not necessarily true that $\mathcal P(\kappa)/I$ is complete for an ideal $I$. In particular if we consider $I=J_{bd}$ t …
- 39.9k
4
votes
How "strong" is the existence of a non trivial ultrafilter on $\omega$?
First of all, not only the existence of a free ultrafilter on $\omega$ is far weaker than $\sf BPI$, even the statement that every filter on $\omega$ can be extended to an ultrafilter on $\omega$ is s …
- 39.9k