On a finite subsemigroup of semigroups in which x^r=x

TL;DR

This paper investigates the structure of semigroups where each element satisfies x^r=x, aiming to determine the minimal size n such that all sufficiently large semigroups with this property contain a subsemigroup of size n.

Contribution

It characterizes the minimal size n for which all large semigroups with the property x^r=x contain a subsemigroup of size n.

Findings

Identifies all such n for given r

Provides structural insights into these semigroups

Establishes bounds on subsemigroup sizes

Abstract

In this paper, we consider the following problem: For every positive integer $r \geq 2$, find all positive integers $n$ such that for every semigroup of order $\geq n$ in which $x^r=x$ for every element $x$ has a subsemigroup of order $n$.