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 $\omega$

TL;DR

This paper characterizes group congruences on a specific semigroup constructed from inductive subsets of ω, showing that such congruences relate to the bicyclic semigroup and describing all non-trivial homomorphic retracts.

Contribution

It provides a complete characterization of group congruences and describes all non-trivial homomorphic retracts of the semigroup based on inductive families of subsets of ω.

Findings

A congruence is a group congruence iff its restriction on the bicyclic subsemigroup is non-trivial.

All non-trivial homomorphic retracts are explicitly described.

The structure of the semigroup's isomorphisms is fully characterized.

Abstract

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.