Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a geometrical notion of dimension for partially ordered sets   The well-known notion of dimension for partial orders by Dushnik and Miller
allows to quantify the degree of incomparability and, thus, is regarded as a
measure of complexity for partial orders. However, despite its usefulness, its
definition is somewhat disconnected from the geometrical idea of dimension,
where, essentially, the number of dimensions indicates how many real lines are
required to represent the underlying partially ordered set.
  Here, we introduce a new notion of dimension for partial orders called Debreu
dimension, a variation of the Dushnik-Miller dimension that is closer to
geometry. Instead of arbitrary linear extensions as considered by the
Dushnik-Miller dimension, we consider specifically Debreu separable linear
extensions. Our main result is that the Debreu dimension is countable if and
only if countable multi-utilities exist. Importantly, unlike the classical
results concerning linear extensions like the Szpilrajn extension theorem, we
avoid using the axiom of choice. Instead, we rely on countability restrictions
to sequentially construct extensions which, in the limit, yield Debreu
separable linear extensions.

26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga