For questions about vector lattices (aka. Riesz spaces; a type of vector space equipped with a partial order), Banach lattices and similar topics.

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A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
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1answer
27 views

Is there a connection between lattices in the sense of orders and lattices in the sense of groups?

I'm wondering about this for some time now - is there a intuitive connection between those concepts or have they been named the same by chance? In particular, my interest in this question was ...
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21 views

6Sz as the automorphism group of the complex Leech lattice

Consider the Leech lattice as a complex lattice over the Eisenstein integers. Both in Conway ("Sphere packings, lattices and groups") and Wilson ("The Complex Leech Lattice and Maximal Subgroups of ...
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16 views

invariance of angles under lattice automorphisms

Let $L$ be a lattice, $x,y\in L$ and $g\in Aut(L)$. Does this imply $$<x,y>=<g(x),g(y)>,$$ in other words, are angles or inner products between any two lattice vectors invariant under ...
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1answer
27 views

Question about an extended linear functional on vector lattice of functions

I'm struggling with following problem: Let $\mathcal{F}$ be a vector lattice of bounded functions on a set $X$ such that $1\in\mathcal{F}$. Suppose that we are given a linear functional $L$ on ...
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1answer
15 views

Sacle the distance of lattice points

I know that for a hexagonal lattice generated by (0,1) and ($\sqrt{3}/2$,1/2) (i.e., when the distance between lattice points is 1), the number of lattice points in a circle of given radius $r$ can be ...
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1answer
20 views

How to find almost periodic lattices in a set of high-dimensional points?

sorry for lame question, but I just have no maturity in this direction. Let's say I have very large set ( millions ) of high-dimensional vectors ( typical dimensionality is 64). These vectors ...
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28 views

What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
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10 views

The size of the vertex hull of a lattice

Let $L$ be a lattice defined by $d$ vectors in $\mathbb{R}^d$. The Vertex Hull of a node $a$ at radius $r$ (denoted $VH(a, r)$) is defined to be the set of vertices that define the convex hull of $L ...
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19 views

inverse relation in a vector lattice

Let $X$ be a vector lattice endowed with an ordering $\leq$. If for some operator $\phi:X_+\rightarrow X$ and $m,M\in X$, we have $$m\leq \phi(x)\leq M,\quad x\in X_+. $$ What would be the bound for ...
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23 views

blocking set of projective planes

Have a quick question, considering the Desarguesian projective plane of order q, what is an upper bound of the minimal blocking set? Wikipedia says the size of the blocking set is bounded below by ...
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1answer
26 views

If $X$ is completely regular and $C(X)$ is a Riesz space (vector lattice) with a unit $u$, is $X$ compact?

Let $X$ be a completely regular space, and let $C(X)$ be a Riesz space (vector lattice) with a unit $u$. Is $X$ compact? The (sort of) converse is true: We know that if $X$ is compact, then ...
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1answer
28 views

Error in my exercise concerning Riesz spaces and Yosida's Lemma.

I was given this exercise in class today: Using Yosida's Lemma, proof that there exists a Riesz-homomorphism $\varphi: BC(\mathbb{R}) \to \mathbb{R}$ such that $\varphi(\textbf{1}) = 1$ and ...
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39 views

Lattice generated by vectors orthogonal to an integer vector

Given a non-zero vector $\boldsymbol{v}$ composed of integers, imagine the set of all non-zero integer vectors $\boldsymbol{u}$, such that $\boldsymbol{u} \cdot \boldsymbol{v} = 0$, i.e., the integer ...
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19 views

Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...
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40 views

Leech Lattice and Golay Code

Consider the following Miracle Octat Generator or MOG. Choose the sign $\pm 3$ and fill in the blanks $\pm 1$ to create a point $x$ in the Leech lattice $\Lambda_{24}$ with $||x||^2=8$. $ ...
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3answers
40 views

Is there a theorem or axiom which shows that permutations of step sequences through a lattice graph result in the same destination?

I have been searching for a theorem, lemma, or even an axiom which shows that the permutations of a step sequence in Taxicab Geometry result in the same destination in such a lattice graph. To ...
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22 views

$\Delta$ vs. $\delta$ in Lattice Theory

I'm learning about lattices and I'd like to confirm the difference between the density $\Delta$ and another density $\delta$. I would greatly appreciate if someone could confirm, correct, or even ...
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1answer
34 views

What does the notation for a group ${A_n}^*$ mean?

I was on a website that catalouges lattices. I was looking through the alternating group lattices and the dihedral lattices and there were two kinds for each. For example, there was $A_2$ and ...
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22 views

Why changing a vector in a basis for a lattice can lead to a new lattice if $gcd(a_{m+1}, \ldots, a_n) = 1$?

On the book "Lattice basis reduction" by M.R. Bremner published by Taylor & Francis Corollary 1.26. Let $L$ be an $n$-dimensional lattice in $\mathbb{R}^n$ with basis $x_1, x_2, \ldots , x_n$. ...
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On characterization of Riesz homomorphisms on $C(X)$ space

How to prove the following: Let $K$ be an arbitrary topological space and $\pi: C(K)\to\mathbb R$ be a map with $\pi (1) = 1$. If $\pi$ is a algebra homomorphism then it is an Riesz homomorphism.
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1answer
68 views

Counting sum of lattice points

Assume a set $S$ with $|S|$ entries. Indeed, $S$ is the set of lattice points inside a $k$-sphere. Assume $V=S\oplus S$ where $\oplus$ is the Minkowski sum of two sets. Do you know any lower bound on ...
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61 views

How to obtaining the lattice corresponding to an elliptic curve

Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding ...
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1answer
20 views

Jordan Decomposition in a Vector Lattice

I'm trying to prove that $x = x^{+} - x^{-}$ for all $x$ in a vector lattice $E$. The relevant definitions: $E$ is a $\bf{\text{vector lattice}}$ if $E$ is a vector space, as well as a partially ...
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55 views

Continuity of a positive preserving operator between C(X) and C(Y)

I've been struggling with this question in Reed and Simon while I'm prepping for quals. Suppose that $T:C(X)\rightarrow C(Y)$ is a positive operator. Prove that T is continuous and $\Vert ...
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263 views

A function that is both log-supermodular and submodular?

Functions that are log-supermodular and submodular do they have some special name? Or any special properties? Any references? Is the function below log-super-modular and sub-modular? Consider ...
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37 views

Is the Voronoi region of a lattice symmetric around origion?

Assume an n-dimensional lattice. Is the Voronoi region of the lattice symmetric around origion? In other word, is the following statement true? "if $x\in \mathcal{V}$ then $-x\in \mathcal{V}$" where ...
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220 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
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1answer
90 views

Different definitions of Banach lattices?

Do I get it correctly, that the are different definitions of "Banach lattices" available in the literature? To be precise, some authors (like Schaefer) include the order continuity of the norm, while ...
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53 views

Computation of determinant of a lattice

Define the following lattice in $\mathbb{Z}^2,$ $$G:=\{\mathbf{x} \in \mathbb{Z}^2:\exists \lambda \in \mathbb{Z} \ \text{such that} \ \mathbb{x}\equiv \lambda (1,1) \mod 5\}. $$ What is the ...
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1answer
53 views

do you know any example which is not lattice norm?

A norm $||.||$ on a Riesz space is said to be lattice norm whenever $|x|\leq |y|$ implies $||x||\leq ||y||$. Do you know any example which is not a lattice norm?
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865 views

Properties of the Cone of Positive Semidefinite Matrices

The set of positive semidefinite symmetric real matrices form 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 ...
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1answer
127 views

Planar cross section of leech lattice?

I was wondering what any planar cross sections of the leech lattice would look like. I don't know much about this topic at all, I'm just quite curious. Is there any way to find out?
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1answer
111 views

How to `bound' $L^\infty$ by the constant function $1$

Let $(S,\mathcal A, \mu)$ be a measure space and consider the Riesz space $L^\infty=L^\infty(S,\mathcal A, \mu)$ (under point-wise ordering). Let $1_X$ denote the indicator function on $S$ (which is ...
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A question regarding vector spaces with partial order

$\newcommand{\N}{\mathbf{N}}$$\newcommand{\R}{\mathbf{R}}$ Let $X=(X,\leq)$ be a Riesz space (a lattice which is also an ordered vector space over $\R$). Define $X^+=\{x\in X\colon x\ge 0\}$. Are the ...
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71 views

Are order continuous Banach lattices with weak unit weakly sequentially complete?

Let $E$ be an order continuous Banach lattice with a weak unit represented as $L^\infty(\mu) \subseteq E\subseteq L^1(\mu)$. Is it true that $E$ is also a weakly sequentially complete Banach lattice? ...
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2answers
137 views

completeness of cones in an ordered normed space

Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
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1answer
292 views

Show sequence equicontinuous

I don't know how to prove this question: Let $X$ be a compact space, and let $(T_{n})$ be a sequence of positive linear operators on $C(X)$. Also, let $f \in C(X)$ be a strictly positive function. ...
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368 views

C*-algebras as Banach lattices?

It seems to be trivial but I am not sure about monotonicity of the norm in the non-commutative case: Is every C*-algebra a Banach lattice with respect to its natural positive cone?
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257 views

How does positivity affect operator norms?

Take a measure space $\Omega$, an exponent $1 \le p < \infty$ and a linear operator $T \colon L^p(\Omega) \to L^p(\Omega)$. Suppose that $T$ is positive, that is $Tf \ge 0$ a.e. if $f \ge 0$ almost ...