Questions tagged [lattice-orders]
Lattices are partially ordered sets such that a least upper bound and a greatest lower bound can be found for any subset consisting two elements. Lattice theory is an important subfield of order theory.
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There are more sublocales than subspaces.
In this article: https://arxiv.org/pdf/2406.12486v1 it is written:
Given a topological space $X$, there are typically more
sublocales in its frame of opens $\Omega(X)$ than subspaces in X.
Can you ...
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How to show that a join of meets equals a meet of joins in a distributive lattice?
On page 30 of Birkhoff's Lattice Theory, Lemma 3 states that in distributive lattices
\begin{gather*}
\bigvee_{\alpha\in A}\left\{\bigwedge_{S_\alpha}x_i\right\}=\bigwedge_{\delta\in D}\left\{\bigvee_{...
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Extending the $M_3,N_5$ theorem from distributive lattices to frames
It is known that a lattice $L$ is distributive if and only if it does not contain the diamond $M_3$ or the pentagon $N_5$ as sublattices.
A complete lattice is one in which every subset has an infimum ...
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If $(y_\lambda)_{\lambda \in \Lambda}$ is a subnet of $(x_\alpha)_{\alpha \in A}$ and if $A$ is a complete lattice, is $\Lambda$ a complete lattice?
Let $X$ be a set, $(x_\alpha)_{\alpha \in A}$ be a net in $X$ and $(y_\lambda)_{\lambda \in \Lambda}$ be a subnet of $(x_\alpha)_{\alpha \in A}$.
If $A$ is a complete lattice (i.e. not only a directed ...
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Is the following object a lattice?
I'd like to check that the following object is a lattice, since I'm having trouble understanding what I read as a slight ambiguity in the definition I've got in front of me.
The set $\{\emptyset,\{1\},...
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Example of a residuated prelinear lattice that isn't linear
A residuated lattice is an algebra
$$(L,\land, \lor, \star,\Rightarrow,0,1)$$
with four binary operations and two constants such that
$(L,\land,\lor,0,1)$ is a lattice with the largest element 1 and ...
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Congruences on the pentagon lattice $\mathcal{N}_5$
Let $\mathcal{N}_5$ refer to the Pentagon lattice, or the lattice generated by the set $\{0, a, b, c, 1\}$ subject to $1 > a$, $1 > c$, $a > b$, $b > 0$ and $c > 0$.
My aim is to find ...
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Lattice with supermodular height function is lower semimodular
Question
Let $(L,\leq)$ be a lattice of finite length and let its height function $h$ be supermodular, meaning that
$$h(x \wedge y) + h(x \vee y) \geq h(x) + h(y) \quad \forall x,y\in L.$$
Does it ...
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How to get the distributive law for an l-group?
In Birkhoff an l-group G is defined as a group that is also a poset and in which group translation is isotone:
\begin{gather*}
x\leq y\implies a+x+b\leq a+y+b\;\forall a,x,y,b\in G,
\end{gather*}
and ...
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The functor $Sob \to Loc $ is faithful
I want to show that the functor $O: Sob \to Loc$ from the sober topological spaces to the locales is faithful. So take $X,Y$ two sober topological spaces. I want to show that the map
$$O_{X,Y}: Hom(X,...
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How to get the height function for modular lattices?
In these notes, it is said that for modular lattices of finite lengths the height function
\begin{gather*}
h(x)=lub\{l(C):C=\{x_0,...,x_n:x_0=O\prec...\prec x_n=x\}\}
\end{gather*}
obeys
\begin{gather*...
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$X\setminus \overline{\{ x\}}$ is meet irreducible in the lattice $O(X)$.
Let $X$ be a topological space and $x\in X$ I want to show that $X\setminus \ \overline{\{ x\}}$ is meet-irreducible in the lattice $O(X)$ of open sets of $X$. This means that we want to show that $X\...
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The open sets form a complete lattice [duplicate]
Let $X$ be a topological space and denote $O(X)$ the set of open sets of $X$. Then I read that
"$O(X)$ is a complete lattice since the union of any family of open sets is again
open". I don'...
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Example of an infinite compact measurable space
Let $X$ be a nonempty set with a $\sigma$-algebra $\mathcal{A}$. The notion of $\sigma$-algebra strictly lies between Boolean algebras and complete Boolean algebras. Clearly, $\mathcal{A}$ is a ...
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Alternative characterization of distributive lattice
Let $(X,{\leq},{\wedge},{\vee})$ be a lattice.
The lattice is called distributive if for all $x,y,z\in X$ both distributive laws hold:
$$
x \wedge (y \vee z) = (x \wedge z) \vee (y \wedge z)
\quad\...
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