**7**

votes

**2**answers

525 views

### Maximal number of edges and triangular cells for n points in a triangular lattice

Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...

**1**

vote

**1**answer

208 views

### Representing integer points inside a polytope using a unit hypercube

Let $D\subset\mathbb{Z}\times\mathbb{Z}$ such that $D$ is formed by the points bounded by a convex polytope. Let $f:D\to\mathbb{R}$ defined by $f(x,y)=ax+by+c$, where $a,b,c\in\mathbb{R}$ and ...

**5**

votes

**2**answers

521 views

### Lattices in SOL

Consider a semi-direct product $\mathbb{Z}^2\rtimes_A\mathbb{Z}$, where $A\in SL_2(\mathbb{Z})$ and $|Tr(A)|>2$. It is clear that it is isomorphic to a lattice in the 3-dimensional solvable Lie ...

**8**

votes

**1**answer

671 views

### The Number of Short Vectors in a Lattice

Given a lattice $L = \bigoplus_{i=1}^{m} \mathbb{Z}v_i$ (the $v_i$ are linearly independent vectors in $\mathbb{R}^n$) and a number $c > 0$, can one quickly compute or find a good estimate on the ...

**4**

votes

**1**answer

443 views

### Lorentzian characterization of genus

Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973)
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right),$$
...

**17**

votes

**3**answers

2k views

### 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 ...

**6**

votes

**2**answers

416 views

### Is the square of the covering radius of an integral lattice/quadratic form always rational?

This is one of many observations from Pete L. Clark's questions on "Euclidean" quadratic forms. I sent Pete many positive integral forms that obeyed his condition. In turn, his condition turns out to ...

**1**

vote

**1**answer

399 views

### Scott topology, but for graphs

Hi,
Would it be possible to define an order theoretic topology on graphs? I am thinking about the scott topology. There would be an associated continuous map on graphs.

**5**

votes

**1**answer

287 views

### Orthogonal Complements of Root Lattices in E_8

I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be primitively embedded in the $ E_8 $ lattice. Is the orthogonal complement of $ L$ in $E_8$ unique up to ...

**0**

votes

**1**answer

543 views

### Neron-Severi Lattice of Elliptic K3

I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass ...

**0**

votes

**2**answers

471 views

### Lattice generators

I hope this question isn't trivial. Let $L$ be a lattice in $\mathbb{C}$ generated by two complex numbers $w_1,w_2$ which are linearly independent over $\mathbb{R}$. Let $\gamma\in\mathbb{C}$ be a ...

**15**

votes

**1**answer

696 views

### Codes, lattices, vertex operator algebras

At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention ...

**0**

votes

**0**answers

173 views

### Ordering labellings of a fixed poset.

Let $\{A_1,\ldots, A_m\}$ be a family of sets and $I=\{1, \ldots, m\}$. Assume for any $J\subset I$, $B_J=\bigcap_{i\in J}A_j$ satisfies $1\leq |B_J| \leq m-1$ as long as $|J|>1$.
We define a ...

**4**

votes

**1**answer

334 views

### Integrating the multinomial over a hypercube

I have come across an integral of the form
$$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i\right)^mdx_1d x_2\dots dx_n.$$
I have a solution that makes use of the partition function, but I ...

**2**

votes

**1**answer

471 views

### Even lattices and binary codes

I have a maybe simple question about even positive-definite lattices and lattices coming from binary codes. They seemed to be used in framed vertex operator algebras.
What is known about even ...

**12**

votes

**2**answers

1k views

### How is the Ising model an example of a lattice model as per Kontsevich?

In section 3.2 of Kontsevich's very interesting paper "Notes on motives in finite characteristic,", he gives an axiomatic definition of a "lattice model" attached to a Boltzmann datum (V_1,V_2,R), ...

**4**

votes

**2**answers

515 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 ...

**7**

votes

**2**answers

525 views

### What groups have a second maximal subgroup below exactly four maximal subgroups?

I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal ...

**2**

votes

**2**answers

177 views

### how to find vertex of parallelotope closest to given point P in R^n ? (Or minimize quadratic form over {+-1}) Is it NP ?

Consider a parallelotope in R^n and some point "P" in R^n.
What algorithms (except of brute force) can be suggested to find the closest vertex of paralleloptope to "P" ?
Is it NP ?
Parallelotope ...

**1**

vote

**0**answers

157 views

### How to find closest point to restricted lattice on the plane ? ( m*h1 + n*h2, for 0<m,n<N)

Consider finite piece of lattice i.e. points of the form m*h1 + n*h2, for 0<m,n<N h1, h2 -some vectors. Consider some point "P" on the plane. How to find ...

**3**

votes

**2**answers

527 views

### Geometric interpretation of distributivity

I'm studying distributivity in the lattice theory, consequently, I'm after any ideas that
might help to develop some intuition.
In elementary school level algebra, distributivity property
...

**9**

votes

**6**answers

555 views

### Which finite nonabelian groups have long chains of subgroups as intervals in their subgroup lattice?

Given N, what is a finite non-abelian (and preferably non-solvable) group G in whose subgroup lattice Sub[G] there is an interval that is a chain of length at least N?
Since N can be arbitrarily ...

**4**

votes

**1**answer

1k views

### Closest vector problem (=nearest lattice point) is trivial for “reduced lattice” ?

Consider some lattice in R^n. Take some point "P" in R^n (which does not belong to this lattice in general). The problem is to find "nearest" lattice point. The problem is known NP-hard in general it ...

**9**

votes

**1**answer

473 views

### Niemeier lattices and theta functions

I have an extremely elementary question. Let's say someone randomly hands you a theta function associated to a Niemeier lattice (unimodular even, n=24). What can you say about which Niemeier lattice ...

**3**

votes

**1**answer

257 views

### How “often” does LLL-reduction produce “optimal” result ? Is there condition (or informal understanding) on lattice that it LLL is optimal ?

Consider some lattice in R^4 (C^4) or C^8.
Famous "lattice reduction" procedures (like LLL latice reduction)
produces some "reduced basis". However in general there results are not "the best ...

**5**

votes

**2**answers

514 views

### Lattice reduction in R^3 (R^4) or what is fundamental domain for SL(3,Z) , (SL(4,Z)) ?

Consider a lattice in R^3.
Is the some "canonical" way or ways to choose basis in it ?
I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|.
Considering lattices with ...

**7**

votes

**2**answers

1k views

### How to find nearest lattice point to given point in R^n ? Is it NP ?

Consider some lattice in R^n.
Take some point "P" in R^n (which does not belong to this lattice in general).
What are the algorithms to find some nearest lattice point to "P" ?
"Nearest" - means in ...

**1**

vote

**2**answers

300 views

### compact elements and continuous functors

Hi,
I am interested in abstracting the Scott topology from Domains to Categories. One can find a definition of a continuous functor which is just such an abstraction:
A functor F:C→D is continuous ...

**6**

votes

**2**answers

452 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 ...

**0**

votes

**0**answers

133 views

### Bogus(?) proof for equivalence of normal and dual distributivity conditions in lattices

I am attempting to prove the equivalence of the following two definitions of distributive lattices:
$(a \lor b) \land c = (a \land c) \lor (b \land c)$
$(a \land b) \lor c = (a \lor c) \land (b \lor ...

**4**

votes

**1**answer

243 views

### Turing Machine which generates order on the set of its states

This question is related to this one Do Turing Machines generates any nontrivial lattice on the set o symbols or states?
The Turing machine (TM) is an abstract model for effective implementation of ...

**8**

votes

**3**answers

1k views

### Listing lattice points in a simplex

Let $n \in \mathbf{Z}_{\geq 1}$. Is there an algorithm which, given a simplex $\Delta \subset \mathbf{R}^n$ specified as the convex hull of $v_0,\dots,v_n \in \mathbf{Z}^n$, computes the set $\Delta ...

**1**

vote

**1**answer

211 views

### Do Turing Machines generates any nontrivial lattice on the set o symbols or states?

Second question, probably better: Turing Machine which generates order on the set of its states
I would like to ask ( if it is not terribly obviously wrong):
Do Turing Machine generates ...

**0**

votes

**0**answers

170 views

### what is bottom in a dcpo of groups

Hi,
Given a dcpo of finitely presented groups, I wanted to say that the free group is bottom, but I don't think that is right. Can anyone say what is a typical bottom in a dcpo of groups? ...

**1**

vote

**1**answer

240 views

### Distributive lattices arising from a collection of sets closed under intersection.

Hello everyone,
It's well known that every collection of subsets of a set X which is closed both under intersection and reunion is also a distributive lattice (the order relation being sets inclusion ...

**4**

votes

**0**answers

211 views

### Domination in Nice Lattices

Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...

**4**

votes

**3**answers

809 views

### Lattices: why require bilinear form to be integral?

This is a quite localized question, but I hope it won't be closed as unfit to MO. Well, a lattice $\Lambda$ in $\mathbb{R}^n$ is a discrete subgroup generated by a basis. Such a lattice gets a ...

**0**

votes

**2**answers

351 views

### Semilattices with n elements

How many n-element semilattices there are?
For example, for n-element partially ordered set we can figured out, that there are $2^{n*(n-1)}$ possible sets.
And can I find all possible n-element ...

**4**

votes

**2**answers

571 views

### additive functions on a lattice

Given a lattice $L$. Can one classify all functions $f:L\rightarrow \mathbb{R}$, that satisfy
$f(a \wedge b)+f(a\vee b) = f(a)+f(b)$.
Some examples are the the set of all finite subsets of a given ...

**8**

votes

**3**answers

633 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 ...

**5**

votes

**4**answers

603 views

### Angles in an integral lattice

Let $d\geq 1$ be a fixed integer, and $\mathbb{Z}^d$ be the lattice of all integers; consider the set $A_d \subset [-1,1]$ defined by
$$ \alpha \in A_d \ \iff \ \alpha=\frac{v \cdot w}{\|v\| \|w\|} ...

**2**

votes

**0**answers

139 views

### Can lattice join be an element of residuated pair?

Given a classic (not Residuated) lattice, with standard definition of partial order via lattice join and meet operations, is it possible to satisfy Galois equivalence
...

**12**

votes

**1**answer

1k views

### Sublattices of Young's Lattice

Young's Lattice is the lattice of inclusions of Young tableaux, or integer partitions.
In R. Stanley's Enumerative Combinatorics Vol. 1, Proposition 1.4.4., there is a generalization of integer ...

**5**

votes

**1**answer

362 views

### Does “antichain” mean something different in set-forcing than in lattice theory?

On page 3 of Introduction to Lattices and Order, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise incomparable elements:
The ordered set P is an ...

**1**

vote

**0**answers

389 views

### Complex Lorentzian Leech Lattice and the Bimonster [closed]

I'm reading an excellent paper on the complex Lorentzian Leech Lattice and the bimonster (Tathagata Basak). Instead of using the binary Golay Code, the author uses the ternary Golay code and the ...

**7**

votes

**1**answer

620 views

### Lattice-ordered commutative monoids

By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...

**0**

votes

**1**answer

336 views

### Do filters complementive to a given filter form a complete lattice?

Really I should first ask this question here on MathOverflow and only then post it as an open problem in Open Problem Garden and propose it as a polymath problem. Indeed I did the reverse and now hope ...

**2**

votes

**1**answer

278 views

### Count of lattices on finite set

Let $p(n)$ denote count of lattices on finite set $G$, $|G|=n$ (without isomorphism). It's know closed formula for $p(n)$?
It's clear, that $1 \leq p(n)$ and also that $p(n-1) \leq p(n)$ for $n \geq ...

**3**

votes

**2**answers

236 views

### What do you call a lattice whose meet operation preserves disjointness of subsets?

To make my question more precise and compact (and probably more intuitive), let me define the following:
A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x ...

**7**

votes

**4**answers

1k views

### Online introduction to Lattice Theory?

Apart from J. B Nation's (revised) Notes on Lattice Theory, is there any other (mostly introductory) material on Lattices available online?