Tagged Questions
1
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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?
0
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1answer
39 views
Represent the three element chain as a subdirect product of subdirectly irreducible lattices.
Represent the three element chain as a subdirect product of subdirectly irreducible lattices.
I know the two element chain as either a Boolean algebra or a semilattice,is subdirectly irreducible.In ...
1
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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 ...
0
votes
2answers
78 views
is the following a Boolean Algebra?
Boolean Algebra:
$$D_{30}=\{n:n\mid30\}= \{1,2,3,5,6,10,15,30\}$$
I don't know how to test that this is a boolean algbra (a BA is a distributive lattice with $T,F$ in which every element has a ...
2
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1answer
86 views
Complete Lattice with unique negation
Suppose i have a complete lattice (meet and join exist for any two subsets of the lattice) and for every element of the lattice, there is a unique element that is its complement (hence a Boolean ...
3
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1answer
160 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
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 ...
4
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1answer
153 views
Question about Boolean algebra and ultrafilters
In the following $B$ denotes a Boolean algebra and $\bar{x}$ is the complement of $x$.
In my notes there is the following theorem:
If $U \subset B$ is an ultrafilter on $B$ then for every $x \in B$ ...
3
votes
1answer
251 views
What are the algebras of the double powerset monad?
Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image ...
0
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1answer
88 views
Boolean algebra-Modular lattice
Let $L=\{a,b,c,d,e,f\}$; $P(L)$ is the set of all partitions of $L$, and $\le$ is the order relation on $P(L)$ defined as:
if $r$ and $t$ are relations, then $r\le t$ iff every block in $r$ is a ...
1
vote
1answer
386 views
How to prove that any x in a complemented distributive lattice cannot have two complements?
How can I prove the following statement?
In a complemented lattice, if there exist two complements for any x then the lattice is not distributive.
I thought of showing that, in a complemented ...
0
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1answer
115 views
Lattices - How to prove a simple inequality?
Lattices are kind of new to me and I'm not yet familiar with all of their properties so excuse me if what I'm asking here is extremely basic or easy.
How can I prove the following inequality for a ...
