0
votes
0answers
8 views

Cardinality of a subalgebra of a boolean algebra

Let $X$ be a subset of a Boolean algebra $B$, and let ,$A$ be the subalgebra generated by $X$. Show that, if $X$ is finite, then $|A|$ $\leq$ $2^{2^{|X|}}$ , and that, if X is infinite, then |A| = ...
0
votes
0answers
15 views

A filterbase generating filter F

Show that a non empty subset $X$ of a filter $F$ in $B$ is a base for $F$ iff $X$ generates $F$ and for all $x,y$ $\in$ $X$ $\exists$ $z $ $\in$ $X$ such that $z$ $\leqq$ x $\wedge$ y.
1
vote
1answer
46 views

non-atomic complete Boolean lattice

Is there a Boolean complete lattice that is not atomic?
0
votes
0answers
26 views

A set with a property in a Boolean lattice.

Are there a Boolean lattice $(X,\le)$, $A\subseteq X$ and $b\in X$, such that $\sup A$ exists but $\sup\{a\wedge b|a\in A\}$ does not exist.
1
vote
0answers
27 views

“Optimal Disjoint Decomposition” of a Boolean Lattice Subset?

I am looking for the name (and, possibly, an efficient solution) of the following problem: Given a Boolean lattice $(L, \sqcap, \sqcup)$ with least element $0$, and a finite subset $X \subseteq L$, ...
2
votes
1answer
49 views

Example of a function between boolean lattices that preserves $(\top,\bot,\wedge,\vee)$ but not complements.

Its easy to find boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ such that $f$ preserves both top and bottom elements, as well as binary meets, but not complements. ...
2
votes
1answer
50 views

Is every Boolean algebra a separative partial order?

A partially ordered set $\langle P,\leq\rangle$ is separative iff it satisfies the following condition: \[ \neg x\leq y\Rightarrow\exists z(z\leq x\wedge z\bot y) \] where: \[ x\bot y\iff\neg\exists ...
2
votes
1answer
40 views

Looking for an algebraic structure

I'm looking for the name of algebraic structures (in which the elements are partially ordered) with the following properties: Top element defined, bottom optional; Join defined for all elements, ...
0
votes
1answer
65 views

Condition for the boolean algebra of clopen sets to be extremely disconnected.

Let $X$ be a topological space and let $\Gamma \mathcal O(X)$ be it's boolean algebra of clopen subsets. For compact totally disconnected space, show that $\Gamma \mathcal O(X)$ is complete (as a ...
3
votes
1answer
60 views

Do all equational theorems of Boolean algebra not involving complementation also hold for all bounded distributive lattices?

Or we might ask the question in the negative: Do there exist equational theorems of Boolean algebra involving only the operations $\wedge,\vee$ and the constants $\top$ and $\bot$ that fail to be ...
3
votes
3answers
145 views

Existence of maximal boolean-algebra sublattice (preserving top and bottom) of finite distributive lattice

If I regard a modal logic as some sort of many-valued logic, a "modal operator" projecting into a classical propositional logic context could sometimes be useful. Such an operator would provide a ...
0
votes
0answers
59 views

Can a brouwerian lattice be extended into a boolean algebra?

Can an arbitrary brouwerian lattice (=locale = frame) be extended into a boolean algebra? What do I mean by "extended"? I don't know. All I know is that our brouwerian lattice is a sub-poset of the ...
1
vote
1answer
83 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
votes
1answer
75 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
vote
1answer
93 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
165 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
votes
1answer
111 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
votes
1answer
276 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?
4
votes
2answers
148 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
votes
1answer
185 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 $x \in B$ ...
5
votes
1answer
361 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
votes
1answer
112 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
748 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 ...
1
vote
1answer
136 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 ...