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Questions tagged [vector-lattices]

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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Why does the lattice need to be even?

I'm studying root systems and lattices in the context of Lie algebra. In Prof. Kac's notes (https://math.mit.edu/classes/18.745/Notes/Lecture_16_Notes.pdf) it states that if $Q$ is an even (in ...
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Minkowski's Lattice Point Theorem on Linear Forms

I am reading Neukirch's Algebraic Number Theory and I was stuck at an exercise. Here is the exercise: Let $$L_i(x_1,\cdots,x_n)=\sum_{j=1}^na_{ij}x_j,\hspace{0.5cm}i=1,2,\cdots,n$$ be real linear ...
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Confusion over the use of the term "module" in mathematics and post-quantum cryptography.

In post-quantum cryptography, there's a suite of algorithms based on "modular lattice". These schemes are defined in terms of vectors and matrices whose elements are polynomials of a fixed ...
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Show that $f_n$ is an increasing $e$-uniform Cauchy sequence

We have $f(\theta )= |x \cos \theta +y \sin \theta | $ and $e=|x|+|y|$ and define for $n\in\mathbb{Z}_{\ge 0}$ $$f_n= \sup \{f(2\pi k 2^{-n}): k=0,1,\dots,2^n \}.$$ We will show that $f_n $ is an ...
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What does $\mathbb{Z}_2^3$ mean? [closed]

What does $\mathbb{Z}_2^3$ mean? Is the subscript $2$ a modulo and the superscript $3$ dimensions of each element? I am studying lattice cryptography and set theory and I would like to know the how ...
smith33444's user avatar
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Shortest Vector in a lattice on a Galois Field

I am trying to solve the shortest vector problem, but on a galois field rather than the real numbers. Say, for example, I am working in the field of integers modulo 7, $Z_7$. If I have the following ...
tobysmith156's user avatar
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Generalized boolean algebra structure on connected subset of euclidean space

This is a curiosity question that I've been grappling with as I've been reading more about lattice theory: Is it possible to endow some connected subset of $\mathbb{R}^n$ with a generalized boolean ...
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Existence of the supremum in vector space complexification

I am struggling with understanding the existence of the supremum. $L$ is a relatively uniformly complete Riesz space. $L+iL=\{φ:φ=f+ig, f,g∈L\}$ is a complexification of $L$. Modulus in the vector ...
Ebru Kılıç's user avatar
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A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$. ...
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What is the similarity or difference between the lattices $\mathcal{L}_1$ and $\mathcal{L}_2$?

Consider the $2$-dimensional $\mathbb Q$-vector space $\mathbb Q^2$. Let $\{v,w \}$ be a basis of this vector space. Consider the following lattices \begin{align} &\mathcal{L}_1=\{iv+jw\},~i,j \in ...
MAS's user avatar
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How to show $A$ is subset of the lattice $B$ in the vector space $\mathbb Q^2$?

I am writing two definitions, the $1$st one is a cover in some sense while the $2$nd one is a lattice: Definition 1: If $m,n$ are integers bigger than $1$, then the set $$A=\{(x,y) \in \mathbb Q^{\geq ...
MAS's user avatar
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Norm inequality for Banach Lattices

Let $X$ be a Banach lattice, $\varepsilon>0$ and $x,y\in S_X$. If $\||x|\pm y\|\le1+\varepsilon$, then is it true that $\||x|+|y|\|\le1+f(\varepsilon)$? Where $f$ is some real function satisfying $\...
Stefano Ciaci's user avatar
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Proving a set is a closed lattice cone in B(S) space

Let $S$ be an arbitrary set and $B(S)$ denote the space of all real-valued bounded functions on $S$. Then we know $B(S)$ is a lattice with pointwise maximum or minimum as the lattice operations. We ...
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What is the lattice diagram of the Galois extension $\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$?

Consider the Galois extension $K:=\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$. As it is clear, it is of degree $8$ extension. So the Galois group is a group of order $8$. I ...
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If $A$ is a linear subspace of the Riesz space $E$, then $A=A^{dd}$.

If $A$ is a linear subspace of the Riesz space $E$, then $A=A^{dd}$. Definition. Let $E$ be a Riesz space. The elements $f$ and $g$ in $E$ are said to be disjoint, we write $f\perp g$, if $|f| \wedge |...
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Why are the row space and column space lattices of a boolean matrix inverses of each other?

We define the row/column space of a matrix $A$ to be all possible sums of rows/columns of $A$. The row space ($V(A)$) and column space ($W(A)$) form a lattice. See example matrix and respective row ...
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Let $E$ be a Riesz space. Show that if $f_n \to f$, then $f_{n_k} \to f$ for every subsequence $(f_{n_k}:k=1,2,\ldots)$ such that $n_1<n_2<\cdots$.

Let $E$ be a Riesz space. Show that if $f_n \to f$, then $f_{n_k} \to f$ for every subsequence $(f_{n_k}:k=1,2,\ldots)$ such that $n_1<n_2<\cdots$. The reference/book was I used is "...
lap lapan's user avatar
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Where do the "classical lattices" come from?

I am trying to understand sphere packing problems, and it seems well-known that lattices can be only of certain kinds: the classical types (e.g. $A_n, B_n$, etc.) or the exceptional types (e.g. $E_8$)....
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Pre-dual of a lattice is a lattice?

Let $X$ be a Banach space such that its dual $X'$ is a Banach lattice. Then must $X$ also be a lattice? I know that if $X$ is a lattice, then so is its dual $X'$. However, I was wondering is the ...
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Discrete subset of $\mathbb{R}^n$

Suppose that $M \in \mathbb{R}^{m \times n}$, we define $$ S(M)=\left\{c_{1} \mathbf{v}_{1}+\cdots+c_{m} \mathbf{v}_{m}: c_{i} \in \mathbb{Z}\right\} $$ where $\mathbf{v}_i$ is the i-th row of the ...
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Finite-dimensional vector lattice is a direct sum of one-dimensional vector lattices

Prerequisites Definition 1 Let $\mathcal{L}$ be a non-empty collection of real-valued functions on a set $X$. Then $\mathcal{L}$ is a real vector space iff for all $f,g\in\mathcal{L}$ and $c\in\mathbb{...
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Example of non-unique extension of measure for the pre-integral in the Stone-Daniell Theorem

Prerequisites Definition 1 Let $\mathcal{L}$ be a non-empty collection of real-valued functions on a set $X$. Then $\mathcal{L}$ is a real vector space iff for all $f,g\in\mathcal{L}$ and $c\in\mathbb{...
Junk Warrior's user avatar
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Finding a measure for a pre-integral on a simple vector lattice that is not a Stone vector lattice

Definition 1 Let $\mathcal{L}$ be a non-empty collection of real-valued functions on a set $X$. Then $\mathcal{L}$ is a real vector space iff for all $f,g\in\mathcal{L}$ and $c\in\mathbb{R},cf+g\in\...
Junk Warrior's user avatar
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Necessary condition for a vector space spanned by a single function to be vector lattice

Definition Let $\mathcal{L}$ be a non-empty collection of real-valued functions on a set $X$. Then $\mathcal{L}$ is a real vector space iff for all $f,g\in\mathcal{L}$ and $c\in\mathbb{R},cf+g\in\...
Junk Warrior's user avatar
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Positivity in $C^*$-algebra vs Riesz spaces

If $A$ is a $C^*$-algebra, then a self-adjoint element $x\in A$ is a called positive if $sp(x)\subseteq [0,\infty)$. I know of the following result: There exist positive elements $x_+,x_-\in A$ such ...
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How to tease out the sampling process of a function on the product of Riesz space and sample space?

We have a function $D(Z;\omega):\mathcal{Z}\times\Omega\rightarrow\mathbb{R}$. $\mathcal{Z}$ is Riesz space, $\Omega$ is sample space. For each given $\omega\in\Omega$, $z$ satisfies the partial order ...
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Question about Cornacchia's algorithm for factoring a prime in a complex quadratic order

The paper "Counting points on elliptic curves over finite fields" by René Schoof presents an algorithm for factoring a prime $p$ on a lattice in $\mathbb{C}$. The problem is, I don't get the ...
José's user avatar
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Is there any relation between monotone operators and increasing operators?

If $V$ is some Riesz space (vector lattice for a partial order, say a Banach space), and if an operator $A\colon V \to V'$ is monotone, i.e $$\langle A(u) - A(v), u-v \rangle \geq 0,$$ does it mean ...
MMML's user avatar
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Regularization of $\max(0,\cdot)$ as an operator in a Hilbert space

Let $H$ be a vector lattice for a (partial) ordering $\leq$. Hence $\max(a,b)$ is defined for $a, b \in H$. Where can I find theory regarding the regularization of $\max(0,\cdot)$ as an operator? By ...
BBB's user avatar
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Spanning vectors of simple cubic lattice

The simplest set of vectors spanning a simple cubic lattice is $\textbf{x}$, $\textbf{y}$, $\textbf{z}$, where they are all mutually perpendicular and of the same length. I wanted to find another set ...
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Are Braid groups special cases of lattice endomorphisms?

Considering that Braid groups are defined over linear 1D lattices, I am pondering what would be the multidimensional equivalent of Braid groups, when we consider endomorphisms between 2D lattices, or ...
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Standard terminology for Federer's "lattice of functions"?

In section 2.5.1 of Federer's book "Geometric Measure Theory," given a set $X$, he says "By a lattice of functions on $X$ we mean a set $L$ whose elements are functions mapping $X$ into ...
Quarto Bendir's user avatar
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1 answer
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Examples (or characterization) of conditionally complete vector lattices

Are there examples of conditionally complete vector lattices that are not subsets of measurable functions (with order induced by cone of non-negative functions)? I ask, because there are results in ...
daw's user avatar
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2 votes
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Why the subset $\{1/n:n \in N^*\} \subseteq \mathbb{R}$ is discrete, but $\{0\} \cup \{1/n: n \in \mathbb{N}^*\}$ is not?

Why the subset $\{1/n:n \in N^*\} \subseteq \mathbb{R}$ is discrete, but $\{0\} \cup \{1/n: n \in \mathbb{N}^*\}$ is not? In "Hermite’s Constant and Lattice Algorithms" by Phong Q. Nguyen, ...
BlockchainThomas's user avatar
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1 answer
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Are lattice operations continuous in the Lipschitz norm?

Denote by $Lip_0(X)$ the set of all Lipschitz functions on a metric space $X$ vanishing at some base point $e \in X$. The norm in $Lip_0$ is defined as fololows $$ \|f\|_{Lip_0} := Lip(f), $$ where $...
Yury Korolev's user avatar
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Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
Mark Roelands's user avatar
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1 answer
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Two different definitions of solid subspaces of a Riesz space

In D.H. Fremlin, Topological Riesz Spaces and Measure Theory, [14C], a subset $F$ of a Riesz space $E$ is defined as solid if $x \in F$ whenever there is an $y \in F$ such that $|x| \le |y|$. In G. ...
Logos's user avatar
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Prove that a solid linear subspace of a Riesz space is a Riesz subspace

In D.H. Fremlin, Topological Riesz Spaces and Measure Theory [14F] it is stated that Let $E$ be a Riesz space. A Riesz subspace of $E$ is a linear subspace which is also a sublattice. A solid linear ...
Logos's user avatar
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Positive Basis of a Riesz Space

Does a Riesz space in general always has a positive basis? ae. if $E$ is a Riesz space, can we assume that there exists a set $B\subset E$ such that it is a basis of the vector space $E$ and for ...
Sagi Buchbinder Shadur's user avatar
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What are Voronoi cells for FCC and HCP Lattices lattices in space?

What are Vornoi cells for FCC and HCP Lattices ? (See plot below for FCC and HCP). Is my uderstanding correct that we should get one of the 5 "parallelohedra" ? If so, that seems to contradict ...
Alexander Chervov's user avatar
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General form of elements in the vector lattice (Riesz Space) generated by a vector space

Can any one tell me how the elements of the set $A^{∧∨}$ will look like? I believe they are of the following form: $∨^m_i∧^{m_i}_{k_i} x_{i,{k_i}}$ where $x_{i,{k_i}}$ is in $A$. But in this case I'm ...
Prince Khan's user avatar
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Find a dual lattice basis for lattice $e_1 = \sqrt 2 \hat e_x, \ e_2 = -\frac{1}{\sqrt 2} \hat e_x + \sqrt{\frac{3}{2}}\hat e_y$

Given the basis vector of a lattice $L$: $$e_1 = \sqrt 2 \hat e_x, \ e_2 = -\frac{1}{\sqrt 2} \hat e_x + \sqrt{\frac{3}{2}}\hat e_y$$ I want to find a set of basis vector for the dual lattice $L^*$. ...
Awoo's user avatar
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How to write $(\wedge _j \vee _i x_{i,j})$ $\vee$ $(\wedge _k \vee _l y_{k,l})$ in terms of $\wedge \vee$?

Can anybody tell me if there's any general formulas for the following: If $i, j, k, l$ runs over a finite set, then $(\wedge _j \vee _i x_{i,j})$ $\vee$ $(\wedge _k \vee _l y_{k,l}) = ?$ $(\wedge _j \...
Prince Khan's user avatar
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Making sense of a gradient for solving an ODE

Consider a random lattice where the position of each vertex $i$ is governed by $$ \eta \frac{d \mathbf{x}_i}{dt}=\mathbf{F}_i(t), $$ where $\mathbf{F}_i(t)$ denotes the total force acting on vertex $i$...
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Find vectors $x_1, ... , x_k$ such that $D_n = Int(x_1, ... , x_k)$

The following set of points is a lattice in $\mathbb{R}^n$: $D_n =$ {$(v_1, ... , v_n)| v_1, ... , v_n \in \mathbb{Z}$ and $v_1 + ... + v_n$ is even} Find vectors $x_1,...,x_k$ such that $D_n = Int(...
flutterbug98's user avatar
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Surjectivity from a order unit space $A$ to the affine function space on the state space of $A$?

In the proof for the theorem in the picture snipped off the book State Spaces of Operator Algebras by Erik M. Alfsen & Frederic W. Shultz, where is the Hahn-Banach Theorem is applied? As far as I ...
C. Ding's user avatar
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1 vote
1 answer
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Stone-Weierstrass Theorem (Lattices)

I am struggling with a portion of a proof concerning the lattice version of the Stone-Weierstrass theorem. In particular, there is a subset $\mathcal{A}$ of the set of all real-valued continuous ...
YL-Wint's user avatar
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2 votes
1 answer
483 views

Range of Riesz projection

Is the range of Reisz projection of an isolated eigenvalue $\lambda$ of operator T over Banach space X equal to $\cup_{k\geq1}\ker(T-\lambda)^k$? It is not hard to show it when X is finite dimensional ...
Hamid Enki's user avatar
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1 answer
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Does the order induced by a self-dual cone produce a Riesz space?

Let $X$ be a Hilbert space, $K$ a closed, convex, and self-dual cone. The latter property means $$ K = \{ y\in X : \ \langle x,y\rangle \ge0 \ \forall x\in K\}. $$ Then $K$ induces an order on $X$ by $...
daw's user avatar
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How to demonstrate the finite height of a lattice?

I would like to ask you for help with a formal demonstration concerning the finite height of a lattice. My lattice is defined like this: is a lattice of vectors, each with exactly $n$ cells. In each ...
claudioz's user avatar
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