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.

Lattices are partially ordered sets such that for any two elements $x$, $y$ there is a supremum $x\vee y$ and infimum $x\wedge y$ of the set $\{x,y\}$. Lattice theory is an important subfield of order theory.

See the Wikipedia entry for more information.

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Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the meet operation. (Join and meet.) The ideals of a…
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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 to know is: "Is the Fibonacci lattice the very best…
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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 partially ordered set can fail to be a lattice.…
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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 that this order does not have the lattice…
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Difference between lattice and complete lattice

Definition of lattice require that any two elements of lattice should have LUB and GLB, while complete lattice extends it to, every subset should have LUB and GLB. But by induction , it is possible to show that if any two elements have LUB and GLB…
chinu
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Can a free complete lattice on three generators exist in $\mathsf{NFU}$?

Also asked at MO. It's a fun exercise to show in $\mathsf{ZF}$ that "the free complete lattice on $3$ generators" doesn't actually exist. The punchline, unsurprisingly, is size: a putative free complete lattice on $3$ generators would surject onto…
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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. There is, however, one necessary and sufficient…
user23211
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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 offer a Bounty in two days, depending on the…
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Free lattice in three generators

By general results for every set $X$ there is a free bounded lattice $L(X)$ on $X$. I would like to understand the element structure of this lattice. The cases $X=\emptyset$, $X=\{x\}$ and $X=\{x,y\}$ are quite easy. But for $X=\{x,y,z\}$ we get an…
Martin Brandenburg
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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 none of its sublattices is isomorphic to $M_3$ or…
Zev Chonoles
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Soberification of a topological space

In Johnstone´s Stone Spaces, he introduces the concept of soberification of a topological space: Let $X$ be a topological space and $\Omega(X)$ the lattice of open subsets of $X$, the soberification of $X$ is $pt(\Omega(X))$, where $pt(A)$ is the…
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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 unique up to isomorphism (if exists)? Or is there a…
tomas.lang
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Are ideals in rings and lattices related?

There are (at least) two notions of ideals: An ideal in a ring is a set closed under addition and multiplication by arbitrary element. An ideal in a lattice is a set closed under taking smaller elements and suprema. They coincide nicely on Boolean…
sdcvvc
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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 map $f^* : \mathscr{P}(Y) \to \mathscr{P}(X)$. By…
Zhen Lin
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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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