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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Galois connections give rise to complete lattices
I am reading Introduction to Lattices and Order, second edition, by Davey and Priestly. On page 161, it says
Every Galois connection $(^\rhd,^\lhd)$ gives rise to a pair of closure operators, $^{\rhd\...
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The supermodularity of probability of intersection [closed]
Given a finite sample space $E$, let $E=\{A_1,A_2,\dots,A_n\}$ be a collection of random events.Then, is $f(S)=\mathbb{P}\{\cap_{A_i\in S}A_i\}$ a supermodular function for $S\subseteq E$?
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If sub-universe $S$ of lattice has congruence $\theta$, does the lattice have a congruence $\lambda = \theta \cap S^2$? [duplicate]
Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. ...
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Understanding the definition of congruences over a lattice
Let $(L, \land, \lor)$ a lattice and $\theta$ a binary relation over $L$. We say $\theta$ is a congruence iff
$$
x_0\theta x_1, y_0 \theta y_1 \Rightarrow (x_0 \lor y_0)\theta(x_1 \lor y_1)
$$
(and ...
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Number of lattices over a finite set
I'm interested in finding a general formula for the number of lattices possible over a finite set $S$ as a function of the set's cardinality.
For instance, how many lattices over $\{1, 2, 3\}$ are ...
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Galois connections for contexts
I am reading Introduction to Lattices and Order, Second Edition by Davey and Priestly.
On page 158, it says that
It can be verified that, for $R\in \mathcal{R}$ and $T\in \mathcal{T}$,
$$R\subset R_{...
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Complete Lattices and the Injectivity of the Restriction $f|_S$ - Verification of Proof
Attempt (General Case)
Conjecture: I want to show that if $X$ and $Y$ are nonempty sets, $(X, \leq)$ is a complete lattice, and $f: X \to Y$ is any well-defined function, then there exists a nonempty ...
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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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