Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On group congruences on the semigroup
  $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in the
  case when a family $\mathscr{F}$ consists of inductive non-empty subsets of
  $\omega$   We study group congruences on the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in the
case when an ${\omega}$-closed family $\mathscr{F}$ which consists of inductive
non-empty subsets of $\omega$. It is proven that a congruence $\mathfrak{C}$ on
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is a group congruence if and only if
its restriction on a subsemigroup of $\boldsymbol{B}_{\omega}^{\mathscr{F}}$,
which is isomorphic to the bicyclic semigroup, is not the identity relation.
Also, all non-trivial homomorphic retracts and isomorphisms of the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ are described.

0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci