All Questions
Tagged with lattice-orders filters
32
questions
2
votes
1
answer
97
views
The equivalences between points in a locale in constructive mathematics
I am currently following the definitions of the book Frames and Locales and those in the articles Pointfree topology and constructive mathematics and Topo-logie. There are at least three ways of ...
- 73
2
votes
1
answer
69
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Boolean algebra's filter as a partially order's filter
This is a basic question I'm trying to figure out: why the Boolean's filter definition corresponds to the order-theoretic definition of filter ?
Here follows the relevant definitions.
Definition 1 (...
- 117
3
votes
1
answer
52
views
Is distributivity of a lattice needed if we want its collection of prime filters to be a Stone space?
Let $L$ be a bounded distributive lattice and let $PF(L)$ denote its set of prime filters.
It is well known that $PF(L)$ is a Stone space if it is equipped with the topology that has sets of the form $...
- 152k
2
votes
1
answer
86
views
Prime filters on a linear ordered lattice
Consider a lattice algebra given by a linear order $(H,\leq)$. We say that $A\subseteq H$ is a prime filter if $\emptyset\neq A\neq H$ and
$x,y\in A$ implies $x\land y\in A$ (where $\land$ is the ...
- 4,813
7
votes
1
answer
142
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How can we identify Ultrafilters that “look” or “were” principal?
Let $X$ be an infinite set, and consider the lattice
$\left(\mathscr P(X), \cap, \cup, \emptyset, X\right)$.
The following is well known:
Lemma Every Ultrafilter is either Principal or contains ...
- 3,375
1
vote
1
answer
33
views
Restricting a filter in a Boolean algebra to a generating set and have it generate a filter
Let $B$ be a Boolean algebra and $S \subseteq B$ be a subset that generates $B$. Is it the case that every filter $x$ of $B$ is equal to the filter generated by $x \cap S$? What if $S$ itself is a ...
- 7,290
1
vote
1
answer
176
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When is every principal filter an intersection of ultrafilters?
The question is in the title: what property does a lattice need to have such that for every element of the lattice $x$, there exists a set of ultrafilters in the lattice such that the intersection of ...
- 4,045
2
votes
1
answer
301
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free ultrafilters on $\mathbb{N}$
Let $\mathbb{N}$ be the set of natural numbers,
I know the fact that the number of free ultrafilters on $\mathbb{N}$ is uncountable.
I have two questions:
1.How to construct these uncountable ...
- 3,203
2
votes
2
answers
61
views
The meet of two filters
Given two filters $\mathcal{F}$ and $\mathcal{G}$ on a set $X$, there's a smallest filter containing $\{F \cup G : F \in \mathcal{F},G \in \mathcal{G}\}.$
(Actually, I can't quite tell if this is a ...
- 68.1k
0
votes
1
answer
71
views
Mapping filters on a subset of some set into filters on that set
Let $\mathfrak{A}$ and $\Gamma$ be boolean lattices with $\Gamma$ being a boolean sublattice of $\mathfrak{A}$.
Let us denote $\mathfrak{F}(X)$ the set of filters on a poset $X$ ordered by set ...
- 5,103
1
vote
2
answers
165
views
Help with a proof that every ultrafilter on a distributive lattice is also prime
Let $(L,\leq,\wedge,\vee)$ be a distributive lattice. Let $F$ be an ultrafilter on $L$, show that if $a,b\in L$ and $a\vee b\in F$ the either $a\in F$ or $b\in F$.
This is how I tackled it, but it ...
- 623
1
vote
2
answers
62
views
Show that $F\cup\{a\wedge x: x\in F\}$ is a filter if $F$ is a filter
Let $(L\leq)$ be a distributive lattice. Let $F$ be a filter on $L$. If $a\in L\setminus F$, show that $G:=F\cup\{a\wedge x:x\in F\}$ is also a filter in $L$
Now $G\neq\emptyset$ as $F\subset G$, and ...
- 623
4
votes
1
answer
241
views
In a finite lattice, every filter is principal.
This proof feels to easy. Suggestions?
PROOF: We note that a filter $F$ is principal iff $\land$$F$ $\in$ $F$. (We define $\land$$F$ = $\land$$\{$$G$: $G$ $\in$ $F$$\}$) Suppose that $L$ is a finite ...
- 690
2
votes
1
answer
138
views
Meet of two elements of a poset, which is a superset of another poset
Let $\mathfrak{A}$ and $\mathfrak{Z}\subseteq\mathfrak{A}$ be two complete lattices (with $\bigcup$ and $\bigcap$ supremum and infimum), order on which agrees.
I will denote $\operatorname{up} a = \{ ...
- 5,103
0
votes
0
answers
95
views
Two complete lattices
Attempting to generalize properties of filter bases in my research:
Let $\mathfrak{A}$ and $\mathfrak{Z}\subseteq\mathfrak{A}$ be two complete lattices (with $\bigcup$ and $\bigcap$ supremum and ...
- 5,103