# Automorphism groups of superextensions of finite monogenic semigroups

## TL;DR

This paper investigates the automorphism groups of superextensions of finite monogenic semigroups, providing explicit descriptions for semigroups with up to five elements, and explores characteristic ideals within these structures.

## Contribution

It offers a detailed analysis of automorphism groups of superextensions of finite monogenic semigroups, including explicit classifications for small semigroups, and examines characteristic ideals.

## Key findings

- Automorphism groups are explicitly described for semigroups of size ≤ 5.
- Superextensions extend associative operations from semigroups.
- Characteristic ideals are characterized within these semigroups.

## Abstract

A family $\mathcal L$ of subsets of a set $X$ is called linked if $A\cap B\ne\emptyset$ for any $A,B\in\mathcal L$. A linked family $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $\lambda(X)$ of $X$ consists of all maximal linked families on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $*: \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study automorphisms of the superextensions of finite monogenic semigroups and characteristic ideals in such semigroups. In particular, we describe the automorphism groups of the superextensions of finite monogenic semigroups of cardinality $\leq 5$.