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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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
28 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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0answers
16 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
29 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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15 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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24 views

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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0answers
14 views

What is the (pre)topology hinted by the Daniell integral

Daniell takes a vector lattice $H$ of a set $\mathbb{R}^X$ which he calls the set of elementary functions. For him, an elementary integral $I$ is a nonnegative functional on $H$ which verifies : if ...
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1answer
52 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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1answer
56 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
17 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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182 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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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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0answers
199 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
72 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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1answer
46 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
46 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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1answer
648 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
120 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
100 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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0answers
108 views

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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68 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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1answer
117 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
262 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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1answer
337 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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1answer
244 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 ...