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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Why are ordered spaces normal? [collecting proofs]
Greets
This is a problem I wanted to solve for a long time, and finally did some days ago. So I want to ask people here at MSE to show as many different answers to this problem as possible. I will ...
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Does the order, lattice of subgroups, and lattice of factor groups, uniquely determine a group up to isomorphism?
If we have a two lattices (partially ordered) - one for subgroups, one for factor groups, and we know order of the group we want to have these subgroup and factor group lattices, is such a group ...
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Simple example of non-arithmetic ring (non-distributive ideal lattice)
Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)?
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$\limsup $ and $\liminf$ of a sequence of subsets relative to a topology
From Wikipedia
if $\{A_n\}$ is a sequence of subsets of a topological space $X$,
then:
$\limsup A_n$, which is also called the outer limit, consists of those
elements which are limits of ...
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The Chinese remainder theorem and distributive lattices
In The Many Lives of Lattice Theory Gian-Carlo Rota says the following.
Necessary and sufficient conditions on a commutative
ring are known that insure the validity
of the Chinese remainder theorem. ...
user23211
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Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$
In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization:
A lattice is distributive if and only if ...
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Introductory text for lattice theory
What would be your recommendation for the text which could be useful for someone starting with lattice theory? Suggestion for both books and online materials/lecture notes are welcome.
This was asked ...
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When a lattice is a lattice of open sets of some topological space?
When a lattice $(L,\leqslant)$ is a lattice of open (or closed) sets of some topological space $(X,\tau)$? Which conditions have to be satisfied? We may assume that $X$ is $T_1$.
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Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?
Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon:
https://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere.
But what I would like ...
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Simplest Example of a Poset that is not a Lattice
A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a ...
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What are the algebras of the double powerset monad?
Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image ...
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Relative chinese remainder theorem and the lattice of ideals
Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
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limsup and liminf of a sequence of subsets of a set
I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$.
It says there are two different ways
to define them, but first gives what is common for the ...
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Show natural numbers ordered by divisibility is a distributive lattice.
I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with $\operatorname{gcd}$ as "$\wedge$" and $\operatorname{lcm}$ as "$\vee$".
I ...
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Properties of the cone of positive semidefinite matrices
The set of positive semidefinite symmetric real matrices forms a cone. We can define an order over the set of matrices by saying $X \geq Y$ if and only if $X - Y$ is positive semidefinite. I suspect ...
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