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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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 ...
