On the group of automorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ with the family $\mathscr{F}$ of inductive nonempty subsets of $\omega$

TL;DR

This paper characterizes the automorphism group of a specific semigroup constructed from inductive subsets of natural numbers, showing it is isomorphic to the integers under addition.

Contribution

It establishes that the automorphism group of the semigroup $oldsymbol{B}_{oldsymbol{Z}}^{oldsymbol{ ext{F}}}$ is isomorphic to the additive group of integers, revealing its algebraic structure.

Findings

Automorphism group is isomorphic to the integers.

Semigroup automorphisms correspond to integer shifts.

Provides a complete description of automorphisms for this class of semigroups.

Abstract

We study automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ with the family $\mathscr{F}$ of inductive nonempty subsets of $\omega$ and prove that the group $\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}})$ of automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ is isomorphic to the additive group integers.