Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lattice norms on the unitization of a truncated normed Riesz space   Truncated Riesz spaces was first introduced by Fremlin in the context of
real-valued functions. An appropriate axiomatization of the concept was given
by Ball. Keeping only the first Ball's Axiom (among three) as a definition of
truncated Riesz spaces, the first named author and El Adeb proved that if $E$
is truncated Riesz space then $E\oplus\mathbb{R}$ can be equipped with a
non-standard structure of Riesz space such that $E$ becomes a Riesz subspace of
$E\oplus\mathbb{R}$ and the truncation of $E$ is provided by meet with $1$. In
the present paper, we assume that the truncated Riesz space $E$ has a lattice
norm $\left\Vert .\right\Vert $ and we give a necessary and sufficient
condition for $E\oplus\mathbb{R}$ to have a lattice norm extending $\left\Vert
.\right\Vert $. Moreover, we show that under this condition, the set of all
lattice norms on $E\oplus\mathbb{R}$ extending $\left\Vert .\right\Vert $ has
essentially a largest element $\left\Vert .\right\Vert _{1}$ and a smallest
element $\left\Vert .\right\Vert _{0}$. Also, it turns out that any alternative
lattice norm on $E\oplus\mathbb{R}$ is either equivalent to $\left\Vert
.\right\Vert _{1}$ or equals $\left\Vert .\right\Vert _{0}$. As consequences,
we show that $E\oplus\mathbb{R}$ is a Banach lattice if and only if $E$ is a
Banach lattice and we get a representation's theorem sustained by the celebrate
Kakutani's Representation Theorem.

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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin