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 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 ...
user23211
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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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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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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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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 ...
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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 ...
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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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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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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 $X=\{x,y\}$ ...
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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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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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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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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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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 ...
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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 ...
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