Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Closures of Union-Closed Families   Given a union-closed family $\mathcal{F}$ of subsets of the universe $[n]$,
with $\mathcal{F}$ not equal to the power set of $[n]$, a new subset $A$ can be
added to it such that the resulting family remains union-closed. We construct a
new family $\overline{\mathcal{F}}$ by adding to $\mathcal{F}$ all such $A$'s,
and call this the closure of $\mathcal F$. This paper is dedicated to the study
of various properties of such closures, including characterizing families whose
closures equal the power set of $[n]$, providing a criterion for the existence
of closure roots of such families etc.

8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int