Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the number of subsemigroups of direct products involving the free
  monogenic semigroup   The direct product $\mathbb{N}\times\mathbb{N}$ of two free monogenic
semigroups contains uncountably many pairwise non-isomorphic subdirect
products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$
is a finite semigroup. It contains only countably many pairwise non-isomorphic
subsemigroups if and only if $S$ is a union of groups. And it contains only
countably many pairwise non-isomorphic subdirect products if and only if every
element of $S$ has a relative left- or right identity element.

34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th