On normed semigroups

TL;DR

This paper introduces a new definition of norms on semigroups, investigates their properties, and applies this to show that the matrix semigroup M_n(K) is not topologically regular, providing new insights into semigroup topology.

Contribution

It proposes a novel definition of normed semigroups and explores their properties, leading to a surprising result about the topological regularity of matrix semigroups.

Findings

New definition of a normed semigroup

Proved M_n(K) is not a topological regular semigroup

Established general results on topological regular semigroups

Abstract

The paper begins by exploring the various definitions of norms on semigroups and then presents a new definition of a normed semigroup. The properties of normed semigroups in the new sense are investigated. The new definition of the norm is used to establish a general result on topological regular semigroups which is then used to prove the surprising result that the semigroup M_n(K) is not a topological regular semigroup.