Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the inclusion ideal graph of semigroups   The inclusion ideal graph $\mathcal{I}n(S)$ of a semigroup $S$ is an
undirected simple graph whose vertices are all nontrivial left ideals of $S$
and two distinct left ideals $I, J$ are adjacent if and only if either $I
\subset J$ or $J \subset I$. The purpose of this paper is to study algebraic
properties of the semigroup $S$ as well as graph theoretic properties of
$\mathcal{I}n(S)$. In this paper, we investigate the connectedness of
$\mathcal{I}n(S)$. We show that diameter of $\mathcal{I}n(S)$ is at most $3$ if
it is connected.
  We also obtain a necessary and sufficient condition of $S$ such that the
clique number of $\mathcal{I}n(S)$ is $n$, where $n$ is the number of minimal
left ideals of $S$. Further, various graph invariants of $\mathcal{I}n(S)$ viz.
perfectness, planarity, girth etc. are discussed. For a completely simple
semigroup $S$, we investigate various properties of $\mathcal{I}n(S)$ including
its independence number and matching number. Finally, we obtain the
automorphism group of $\mathcal{I}n(S)$.

27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat