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.$$
Does it ...
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How to get the distributive law for an l-group?
In Birkhoff an l-group G is defined as a group that is also a poset and in which group translation is isotone:
\begin{gather*}
x\leq y\implies a+x+b\leq a+y+b\;\forall a,x,y,b\in G,
\end{gather*}
and ...
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The functor $Sob \to Loc $ is faithful
I want to show that the functor $O: Sob \to Loc$ from the sober topological spaces to the locales is faithful. So take $X,Y$ two sober topological spaces. I want to show that the map
$$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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Möbius function of distributive lattice only takes values $\pm 1$ and $0$.
In this Wikipedia article, I found the statement
[...] shares some properties with distributive lattices: for example, its Möbius function takes on only values 0, 1, −1.
My question is: How it can ...
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A seeming example of a group whose subgroup lattice is lower semimodular but not consistent: where's my error?
Corollary 5.3.12 in Schmidt's "subgroup lattices of groups" states that if groups $A,B$ have lower semimodular subgroup lattices, then so does their direct product $A \times B$.
This paper ...
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What can be said about a Bi-Heyting algebra when the complement operations are useless?
Imagine, for whatever reason, a bounded lattice[1] $(L, 0, 1, ∧, ∨)$ which is a bi-heyting algebra, i.e. there is an operation $→$ such that $x∧y ≤ z$ iff $x ≤ y→z$ (the heyting algebra structure) and ...
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Stone-Cech compactification via lattice ideals of $Coz(X)$
While studying P. T. Johnstone's book Stone Spaces I have come across the following Proposition (Section: 3.3)
Where $max C(X)$ is the set of all maximal ideals of $C(X)$ (ring of real-valued ...
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The join of two set partitions in the refinement order
Let $X$ be a set. The set $\Pi_X$ of all partitions of $X$ is partially ordered via the refinement order, which is defined by $\alpha \leq \beta$ if and only if for each block $A\in \alpha$ there is a ...
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In this proposition why is $\approx$ used rather than $=$?
Proposition 1.1.3 Residuated Structures in Algebra and Logic by Metcalfe, Paoli, and Tsinakis states in part: There exists a bijective correspondence between lattices and algebras $\langle L, \wedge, \...
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Dual of a complete lattice via homset
Suppose $P, Q$ are complete lattices. Define $Hom(P,Q)=\{f\colon P\rightarrow Q\vert f \ \text{preserves arbitrary sups}\}$. Any such $f$ will preserve $0$. And, $P\cong Hom(2,P)$, where $2=\{0,1\}$ ...
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Reconstructing a closure operator from a set of fixed points
Let $L$ be a lattice, not necessarily complete. We define a closure operator as a function $f\colon L\to L$ which is:
idempotent, $f(f(x)) = f(x)$,
isotone, $x\leq y \Rightarrow f(x) \leq f(y)$,
...
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Zorn's lemma: counterexample to chain with upper bound?
The premise required for invoking Zorn's lemma is that every chain in $X$ have an upper bound. So that makes me wonder: what is a good example of a poset $X$ for which that property is false? That is ...
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sigma-algebra vs sigma-frame
A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a family of subsets of $X$ such that:
$\phi\in \mathcal{A}$.
$\mathcal{A}$ is closed under countable unions.
$\mathcal{A}$ is closed under ...
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Congruence lattice of a semiring
A famous result of Funayama and Nakayama states that the congruence lattice of any lattice is a distributive lattice [1]. Also, it can be proved that the lattice is a frame/ complete Heyting algebra 2....
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Partial order where only some elements are reflexive
Are there interesting examples of "almost" partial orders $\preccurlyeq$, where only some elements $x$ satisfy the reflexivity axiom $x \preccurlyeq x$, but every $x$ has at least some $y$ ...
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Is there a connection between the topology of $\mathbb{R}/\mathbb{Q}$ and the logic fundamental to Smooth Infinitesimal Analysis?
As I read through the discussion related to this question (Visualizing quotient groups: $\mathbb{R/Q}$) posted 11 years ago I am reminded of a line on page 20 in a book by J. L. Bell, A Primer of ...
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"No infinite discrete subspace" vs "No infinite pairwise disjoint family of opens"
What is the relationship between the two properties (for a topological space $X$)
$A$: "$X$ has no infinite family of pairwise disjoint open subsets"
$B$: "$X$ has no infinite discrete ...
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Does every set with a supremum contain a monotone net converging to that supremum?
It's well known that if $U \subset \mathbb{R}$ is bounded, then there exists a monotone increasing sequence $(x_{n})^{\infty}_{n=1}$ converging to $sup(U)$.
My question is: Let $X$ be a lattice, and ...
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Finite distributive lattices and finite abelian monoids
A structure of semilattice over T is the same thing than a structure of finite abelian monoid such that $\forall t \in T$, $t² = t$.
Given a semilattice T, we get an abelian monoid by defining $a.b$ =...
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Neutral Elements and Distributive Sublattices from 3 Generators
For neutral elements in a lattice, the definition is typically stated as follows:
An element a of a lattice L is neutral, iff every triple {a, x, y} generates a distributive sublattice of L.
Which now ...
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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 ...
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How to frame the dual statement in a lattice ordered set or an algebraic lattice in general
I am learning the theory of posets and lattices which will eventually lead to Boolean Algebra. I am stuck with the proper understanding of the concept of duality. Followings are what I have gathered ...
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Reference for quotient lattices and universal property?
Question: Are there any references explaining the definition (or definitions) of quotient objects in the category of lattices?
In particular a characterization in terms of universal properties would ...
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Relation between semiring structure of natural numbers and their hereditarily finite sets structure under bitwise operations
Some background
The natural numbers $\mathsf{Nat}=\{0,1,2,\dots\}$ has the structure of a semiring under addition and multiplication.
Write $\mathsf{HFS}$ for the set of all hereditarily finite sets (...
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Lattices/Topology and the Stone Duality
For some context I have some partial understanding of lattices and an intermediate understanding of topology.
I at some point in the past week started thinking about a funny way to view a topology on ...
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Proof for property of derivation operator for formal context: $A_1\subseteq A_2 \implies A^{'}_2\subseteq A^{'}_1$.
I wanted to verify the validity of my proof for the property of derivation operator for formal context.
I will not go to detail that much and simply define few things before indroducing the property ...
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Would the Following Table Strategy Work as an Intuitionistic Decision Procedure?
I had previously sought some insight for handling logical operators in the Rieger-Nishimura lattice and, with assistance here, was able to work out a fairly rigorous way. To the best of my ability, I ...
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Lattice defined on poset vs. Lattice defined on group?
I've seen two different definitions of the term lattice, one is defined on poset, the other one is defined on group. I believe these two are fundamentally different mathematical objects. But I'm not a ...
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Are There Universal Entailments Under the Rieger-Nishimura Lattice for Conditionals When the Antecedent is Higher on It?
I'm working on a bottom-up (atomics-to-proposition) intuitionistic decision procedure, and I encountered some fruits with the Rieger-Nishimura lattice. Specifically, I am looking at this article from ...
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Show that the set of all partitions of a set S with the relation refinement is a lattice.
This one may be one duplicate of QA_1, but its example $\{\{a,d\},\{b,c\}\}\wedge\{\{a\},\{b,c,d\}\}$ seems to not meet the definition in the book because $(\{a,d\} \not\subseteq \{a\}) \wedge (\{a,d\}...
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Lift and frame matroids.
I want to read more about lift matroid and frame matroid and their flats and relations to signed graphs, do you know any basic resources for this?
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