Lattices as they are used in number theory. (Not to be confused with lattice theory or lattices as used in physics!)

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Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
5
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1answer
366 views

coloring in lattice

This is a mathematical question raised from engineering and physics: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
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2answers
160 views

Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups. Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
12
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6answers
2k views

Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here. Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$. Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...
3
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1answer
547 views

Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant. Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
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1answer
789 views

A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
4
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0answers
280 views

Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there. Recall that a subfactor is Dedekind if all its intermediate subfactors are normal. A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
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2answers
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The Gauss circle problem on a hexagonal lattice

Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice ...
2
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1answer
499 views

Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
59
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5answers
3k views

How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
19
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0answers
783 views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
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3answers
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A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one

It turns out that each of Pete L. Clark's "euclidean" quadratic forms, as long as it has coefficients in the rational integers $\mathbb Z$ and is positive, is in a genus containing only one ...
3
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2answers
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An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

In a previous question: (The Gauss circle problem on a hexagonal lattice) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a ...
2
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1answer
346 views

Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they ...
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0answers
190 views

$\mathcal{L}(H_i \subset G_i)$ distributive $\Rightarrow$ $\mathcal{L}(H_1 \times H_2 \subset G_1 \times G_2)$ modular?

Let $\mathcal{L}( G)$ be the lattice of subgroups of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups. Definitions: A lattice $(L, \wedge, \vee)$ is distributive if, ...
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6answers
8k views

Good lattice theory books?

A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...
33
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16answers
3k views

Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
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2answers
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A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
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4answers
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Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
11
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1answer
600 views

Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$ Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...
10
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1answer
646 views

Which lattices have more than one minimal periodic coloring?

The lattice $\mathbb{Z}^n$ has an essentially unique (up to permutation) minimal periodic coloring for all $n$, namely the "checkerboard" 2-coloring. Here a coloring of a lattice $L$ is a coloring of ...
8
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3answers
731 views

Localic locales? Towards very pointless spaces by iterated internalization.

One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames, certain sorts of ...
1
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1answer
132 views

Uniqueness of minimal completions of a partially ordered set

The partially ordered set $(Y,\le)$ is called a superspace of the partially ordered set $(X,\le')$ iff $X\subseteq Y$. $\le'=(\le\cap X^2)$. and a completion of $(X,\le')$ if in addition $~~ 3$. ...
10
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1answer
254 views

Why do the projections in the Calkin algebra not form a lattice?

Let $H$ be an infinite dimensional separable complex Hilbert space. Denote by $\mathcal{B}(H)$ the C*-algebra of bounded operators on $H$, $\mathcal{K}(H)$ the ideal of compact operators on $H$, and ...
10
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3answers
618 views

A “round” lattice with low kissing number?

Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. ...
5
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1answer
175 views

Average of Short Character Sum over All Dirichlet Characters Mod n

Cross-posted from M.SE. Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$. I want to find an ...
2
votes
2answers
490 views

Embedding a brouwerian lattice into a boolean lattice

I have already asked a similar question at http://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra but have received no answer. Sorry, I ask a ...
6
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1answer
396 views

Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$. It is called primitive if it is transitive and preserves no ...
6
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2answers
552 views

Dense sphere packings which are not lattice packings

This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
4
votes
2answers
246 views

Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question. For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...
3
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0answers
272 views

What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$. ...
1
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1answer
236 views

Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors. A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...
10
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3answers
631 views

orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices. QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not necessarily ...
6
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3answers
590 views

Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$

Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of ...
5
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4answers
419 views

Is every Heyting algebra the Lindenbaum algebra of an intuitionistic first order theory?

This question comes after the comments in the recent related question Sigma-complete Lindenbaum algebras?, but in its current form is sufficiently different in my opinion, and so I decided to follow ...
4
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1answer
193 views

Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the ...
4
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2answers
622 views

diameter of Voronoi cell of the lattice ? What about R^n ? What about small n =2,3,4 ?What about random lattice ?

Consider a lattice in R^n. Consider Voronoi cell of it. What is known about diameter ? About the shape ? What are good references ? As far as I understand they are not easy to compute. May be in ...
2
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1answer
420 views

Probability that a “closable” self-avoiding random walk forms a polygon

Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
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0answers
273 views

Determine lattice basis from given lattice points

I'm working on the Shortest Lattice Vector Problem (SVP) for a paper that I'm currently writing. I wish to verify whether a particular structural, namely the building block property ( refer to the ...
0
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1answer
397 views

How to recognize if a lattice is distributive? [closed]

I know that a Boolean lattice must be distributive. But what with these lattices? Are these distributive? $\hskip0.7in$ How to recognize which lattices are distributive or not only by looking on ...