Tagged Questions
1
vote
1answer
51 views
Atomic Boolean lattice is weakly atomic
The book "Introduction to Lattices and Order" says at Exercise 10.12 that an atomic Boolean lattice is weakly atomic.
Could you tell me why it holds?
1
vote
1answer
62 views
The cardinality of finite Boolean lattice
A book (Introduction to Lattices and Order) says that we can prove that the cardinality of any finite Boolean lattice $B$ is a power of 2 by induction on $|B|$ with the following lemma:
If a lattice ...
2
votes
1answer
98 views
Do filters on a Boolean algebra also make a Boolean algebra?
Let $\mathfrak{B}=(B,\bot,\top,\lnot,\wedge,\vee)$ be a boolean algebra. $B_F$ be the set of all filters on $\mathfrak B$. And for all filter $F$, $G$, $F \wedge_{B_F} G \colon= \mathbf C(F \cup G)$ ...
3
votes
1answer
162 views
Cardinality of the set of ultrafilters on an infinite Boolean algebra
Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
3
votes
1answer
124 views
Inducing maps between Boolean completions of posets
Given a separative poset $P$, we can form it's Boolean completion $B(P)$. This is a Boolean algebra whose elements are regular cuts on $P$, as defined here. Also, $P$ embeds densely into $B(P)$ by ...
3
votes
2answers
109 views
Stone duality for ideals and filters (exercise)
In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says:
Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...
