Algebraic Semigroups are Strongly {\pi}-regular

TL;DR

This paper proves that in algebraic semigroups over a field, each element's some power belongs to a subgroup, making the semigroup strongly { ext{ extpi}}-regular, which advances understanding of their algebraic structure.

Contribution

It establishes that algebraic semigroups over a field are strongly { ext{ extpi}}-regular by showing each element's power lies in a subgroup, a novel structural result.

Findings

Existence of a positive integer n such that x^n is in a subgroup for all x

Semigroup S(F) is strongly { ext{ extpi}}-regular

Applicable to algebraic semigroups over any field

Abstract

Let $S$ be an algebraic semigroup (not necessarily linear) defined over a field $F$. We show that there exists a positive integer $n$ such that $x^n$ belongs to a subgroup of $S(F)$ for any $x \in S(F)$. In particular, the semigroup $S(F)$ is strongly {\pi}-regular.