# Tagged Questions

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### Dense Boolean subalgebras

I was reading this page and, in the third part of the first remark I found the definition of dense sub-algebra of a Boolean algebra. It is stated that there are various equivalent definitions of this ...
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### 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| = ...
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### 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.
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### non-atomic complete Boolean lattice

Is there a Boolean complete lattice that is not atomic?
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### 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.
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### “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 $X \subseteq L$, ...
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### 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. ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $x \in B$ ...
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### 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 ...
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### 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 ...