On group congruences on the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ and its homomorphic retracts in the case when a family $\mathcal{F}$ consists of inductive non-empty subsets of $\omega$

Abstract

We study group congruences on the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ and its homomorphic retracts in the case when an $\omega$-closed family $\mathcal{F}$ which consists of inductive non-empty subsets of $\omega$. It is proven that a congruence $\mathfrak{C}$ on ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ is a group congruence if and only if its restriction on a subsemigroup of ${\mathbf{B}}_{\omega }^{\mathcal{F}}$, which is isomorphic to the bicyclic semigroup, is not the identity relation. Also, all non-trivial homomorphic retracts and isomorphisms of the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ are described.

17 pages, in Ukrainian