All Questions
Tagged with lattice-orders vector-lattices
25
questions
1
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0
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70
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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 ...
- 107
0
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0
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37
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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 ...
- 23
3
votes
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38
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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 ...
- 73
1
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27
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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 ...
- 644
2
votes
1
answer
103
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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 ...
- 50.4k
1
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1
answer
68
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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. ...
- 644
1
vote
1
answer
126
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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 ...
- 644
0
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0
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57
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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 ...
- 1,544
1
vote
1
answer
99
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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
$...
- 50.4k
1
vote
1
answer
745
views
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 ...
- 113
0
votes
1
answer
55
views
Math theory that deals with ordered attr-value items?
There is partially ordered sets and lattices.
Is there a branch of math that deals with ORDERED Attribute-Value items/objects.
F.e. av-items /see that attrs also can be missing i.e. doors&roof/ :
...
- 149
0
votes
1
answer
55
views
closed lattice ideal is isomorphic to $C(K)$
Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is ...
- 1,348
1
vote
0
answers
94
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Establish Archimedean property of a vector-lattice
I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem.
I feel the statement below (or in fact weaker versions) should ...
- 11
0
votes
1
answer
189
views
A question about the supremum and infimum in a Banach lattice
Given a Banach lattice $X$, it is well know that every finite subset $A$ of $X$ has a supremum and an infimum.
For example, if $\{ x_1, \ldots, x_n\}$ is a finite subset of $X$, then we have
$$ sup (...
- 871
0
votes
1
answer
48
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Lattice of vectorial space
Let $V$ be a vectorial space and $L(V)$ a lattice of the subspaces of $V$. Show that $L(V)$ is distributive if and only if $dim(V)=1$.
I don't even know how to start this problem. Any hint would be ...
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