Advertising finite commutative semigroups

TL;DR

This paper explores the structure of finite commutative semigroups, providing efficient proofs, detailed analysis of specific examples like Z/nZ, and introducing new results and directions for further research.

Contribution

It offers a comprehensive and efficient structural analysis of finite commutative semigroups, including new results on semigroups defined by generators and relations.

Findings

Structure of Z/nZ depends on prime factorization

Efficient methods for describing semigroup structures

New results on semigroups defined by generators and relations

Abstract

Every mathematician is familiar with the beautiful structure of finite commutative groups. What is less well known is that finite commutative semigroups also have a neat and well-described structure. We prove this in an efficient fashion. We unravel the structural details of many concrete finite commutative semigroups. Here "concrete" comes in two types. First, we examine the structure of the MULTIPLICATIVE semigroups Z/nZ (more interesting than their bland additive siblings) and show that it depends on the prime factors of $n$ in interesting ways. Second, we thoroughly treat finite commutative semigroups defined by generators and relations. Our aim is to provide a comprehensive introduction to the area, including some novel results and some enticing directions for the expert to follow.