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.

110 questions
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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 ...
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Why does dot product give correct angle, but cross product does not?

Question: The primitive lattice vectors for a body-centered cubic lattice are given by $\vec a_1 =\frac{a}{2}(1,1,-1)$ , $\vec a_2 =\frac{a}{2}(-1,1,1)$ , $\vec a_3 =\frac{a}{2}(1,-1,1)$. Calculate ...
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Lattices in $\ell^\infty(X)$

Let $X$ be a set and let $A$ be a subspace of $\ell^\infty(X)$. Prove that for any $f,g \in \ell^\infty(X)$, $f+g = f \vee g + f \wedge g$ and $|f-g| = f \vee g - f \wedge g$. Show that $A$ is a ...
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Vector lattice and sublattice

Consider the integer lattice $(\mathbb{Z}^2,\leq)$ with the canonical partially ordered relation $(a,b)\leq(c,d)$ iff $a\leqslant c$ and $b\leqslant d$. Now just consider a partially ordered subset of ...
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Riesz space of functions from any set $X$ to $\mathbb{R}$

We collect certain functions from $f:X\to\mathbb{R}$ to $\mathscr{T}$ so that it is a vector space over $\mathbb{R}$ under the usual function addition, scalar product. Now we define an partial order ...
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Whom is the decomposition theorem for finite-dimensional Riesz spaces (vector lattices) is attributed to?

There is a result in the theory of Riesz spaces suggesting that every such finite-dimensional space over R, is isomorphic to a direct sum of orthogonal ideals, each one being a lexicographically ...
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Are the two given lattices related to each other by a simple rotation?

I am trying to come up with a general algorithm to determine if two lattices are related to each other by a simple rotation operation. The way I think about a lattice is as an array of points. The ...
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“Representation Capacity” of Finite Lattice Ordered Modules

Apologies for what is probably a very basic question, but I am looking for references for what sort of lattices can be represented in what I think should be called something like "finite lattice-...
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Example of a case that shows that a Hilbert lattice of infinite dimension is not modular.

Let $H$ be a separable Hilbert space. Let $C(H)$ be the set of closed subspaces of the Hilbert space $H$. Let $\le$ be the inclusion $\subseteq$. For $a,b \in C(H)$ define $a^\perp$ as the ...
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Bounded continuous Riesz isomorphisms

I want to know if $C(X)$ and $C(Y)$ are Riesz isomorphic is it true that $BC(X)$ (the space of bounded continuous functions on $X$) and $BC(Y)$ also have to be Riesz isomorphic. I already found the ...