Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On certain semigroups of transformations with an invariant set   Let $X$ be a nonempty set and let $T(X)$ be the full transformation semigroup
on $X$. The main objective of this paper is to study the subsemigroup
$\overline{\Omega}(X, Y)$ of $T(X)$ defined by \[\overline{\Omega}(X, Y) =
\{f\in T(X)\colon Yf = Y\},\] where $Y$ is a fixed nonempty subset of $X$. We
describe regular elements in $\overline{\Omega}(X, Y)$ and show that
$\overline{\Omega}(X, Y)$ is regular if and only if $Y$ is finite. We
characterize unit-regular elements in $\overline{\Omega}(X, Y)$ and prove that
$\overline{\Omega}(X, Y)$ is unit-regular if and only if $X$ is finite. We
characterize Green's relations on $\overline{\Omega}(X, Y)$ and prove that
$\mathcal{D} =\mathcal{J}$ on $\overline{\Omega}(X, Y)$ if and only if $Y$ is
finite. We also determine ideals of $\overline{\Omega}(X, Y)$ and investigate
its kernel. This paper extends several results appeared in the literature.

14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat