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1
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0answers
177 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 $B$. ...
6
votes
1answer
337 views

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 ...
0
votes
0answers
146 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, ...
1
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0answers
102 views

Standard name for a Monoid/Semigroup with a+b <= a, b?

I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice? For instance, for positive reals a, b define a |+| b === 1/((1/a) + (1/b)), you ...
4
votes
1answer
95 views

When are the congruence lattices nicer?

This is a purely idle question, but one I'm increasingly interested the more thought I put into it: For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a ...
6
votes
0answers
114 views

Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying $b$ is bilinear, ...
3
votes
0answers
110 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
1
vote
1answer
189 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 ...
2
votes
0answers
46 views

Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}_n$. Consider the real (positive) convex cone $\mathbb{R}^+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...
5
votes
1answer
307 views

Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$

For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$. It is well known that in dimension ...
5
votes
2answers
146 views

Covolume of the row span of a matrix and of the kernel of a matrix

Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The covolume $\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a fundamental domain for $L$, i.e., the volume of the parallelopiped ...
7
votes
1answer
256 views

Examples of fundamental domains

Everyone knows that it's difficult to compute a general fundamental domain for an arithmetic group but are there any specific examples where such domains have been calculated? I'm mostly interested in ...
8
votes
1answer
180 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 ...
1
vote
1answer
370 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 ...
10
votes
2answers
327 views

Inequalities for averaging over partially ordered sets

Let's start from a classical inequality: If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then $(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$. It can be written also in ...
20
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7answers
1k views

Was lattice theory central to mid-20th century mathematics?

Four years ago, I read a book on the history of mathematics up to 1970 or so. It was very interesting up until the end. The last few chapters, though, were on lattices. The author claimed that ...
0
votes
0answers
40 views

Is it true that the set of points minimizing their distance to a multiset of intervals from a distributive lattice is an interval?

Let $(E, \preceq)$ be a finite distributive lattice, $H_E$ be the Hasse diagram of $E$ and $d$ be the distance on $E \times E$ defined as the length of the shortest path in $H_E$ between any pair of ...
1
vote
0answers
159 views

A question on the poset of classes of isomorphic subgroups of finite groups

Given a finite group $G$, we consider the set $${\rm Iso}(G)=\{[H]\mid H\leq G\},$$where $[H]=\{K\leq G\mid K\cong H\}, \forall H\leq G$. Then ${\rm Iso}(G)$ can be partially ordered by defining ...
29
votes
15answers
2k 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 ...
3
votes
0answers
102 views

Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows: For a ...
22
votes
0answers
366 views

Is there an Ehrhart polynomial for Gaussian integers

Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
2
votes
1answer
618 views

Theta Functions and Cousins

So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...
4
votes
1answer
104 views

Periodic functions over different lattices in $\mathbb R^d$ are linearly independent [closed]

I have the following claim that I think have been proved by someone, but I can not find the reference, hence I would like to ask for help. Here is the claim: Let $f_1, \ldots, f_n$ be continuous ...
0
votes
1answer
80 views

A limit of a sum related to integer lattice and power series

I have the following lemma that I would like to find a source to cite for. Let $L$ be a subset of $\mathbb Z^d_{>0}$. I would like to claim that the limit $$\lim_{z \to (1,\ldots,1)^-} (\sum_{v ...
1
vote
1answer
111 views

generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?
10
votes
3answers
408 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 ...
4
votes
1answer
350 views

Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
1
vote
0answers
56 views

Is this related to a simple property of a lattice?

I am looking for a certain notion of sparseness of lattices. I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. Or at least, I ...
11
votes
1answer
275 views

Defining measures over frames in place of $\sigma$-algebras

Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...
2
votes
0answers
238 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 ...
5
votes
4answers
235 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 ...
7
votes
2answers
522 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 ...
1
vote
1answer
381 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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votes
0answers
97 views

Krein-Rutman version of Hahn-Banach

Consider an arbitrary set of normed Riesz spaces $(X_i,\Vert \cdot \Vert_i,\leq)$, $i\in I$ ($I$ can be compact). Can I apply the Krein-Rutman version ( see Schaefer; TVS; Corollary 2 of 5.4, ...
3
votes
1answer
164 views

Is this bounded from below?

Let $u_1, u_2, u_3 \in \mathbb{Z}$ such that $u_1^2 + u_2^2 = u_3^2$. Is $(u_3 + \frac{u_1 + u_2}{\sqrt{2}})^2$ bounded from below? The irrationality of $\sqrt{2}$ certainly precludes zero, but can ...
2
votes
1answer
99 views

Has the single sorted case of formal concept analysis been investigated?

A formal context in formal concept analysis is a triple $K = (G, M, I)$ where $G$ is a set of objects, $M$ is a set of attributes and the binary relation $I \subset G \times M$ shows which objects ...
7
votes
0answers
177 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, ...
3
votes
1answer
77 views

existence of more distributive Boolean lattices

Are there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that $$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in ...
4
votes
1answer
539 views

Lovász $\delta$ condition for LLL Algorithm

http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm What is the importance of the $\delta$ parameter for LLL bases called Lovász condition? ...
2
votes
4answers
364 views

Minimal (semi)lattice containing a given poset

For a given poset, (I think that) it is easy to construct the minimal join-semilattice containing that poset. I wonder whether the minimal lattice containing that poset is also easy to construct. I ...
2
votes
0answers
277 views

Constructing the Stone Space of a Distributive Lattice

Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian logics"? ...
17
votes
5answers
550 views

Why is the right permutohedron order (aka weak order) on $S_n$ a lattice?

This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the ...
3
votes
0answers
126 views

references for properties/examples of breadth in (semi)lattices

This is in some sense following up on my earlier question and the answer given by NN. I am currently revising the paper which used the condition mentioned in my question. It was pointed out in NN's ...
4
votes
2answers
152 views

Bound on Minimal Length of Vectors in Lattice and its Dual Lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
14
votes
2answers
542 views

Maximal number of maximal subgroups

Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
0
votes
2answers
136 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 ...
2
votes
0answers
173 views

Existence of inclusions of finite groups with a particular lattice property

Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by : $(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...
2
votes
2answers
123 views

Are there atomistic ortholattices which are not modular?

Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise ...
0
votes
0answers
98 views

Siegel's Mean Value Theorem by Rogers and Macbeath

I recently became engaged in the work of Siegel, Schmidt, Rogers, Macbeath regarding random lattices and geometry of numbers, e.g. Siegel proved that $$\int_{SL(n,\mathbb{R})/SL(n,\mathbb{Z})} \sum_{ ...
3
votes
1answer
63 views

Algebraicity of isogenies as maps of lattices

Let $E_i\colon y^2=4x^3+A_ix+B_i$, for $i=1,2$ be two elliptic curves where $A_i,B_i \in \mathbb C$ are algebraic over $\mathbb Q$. For $i=1,2$ let $\Lambda_i\subseteq \mathbb C$ be the unique lattice ...