# 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 ...
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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 ...
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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 ...
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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 ...
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