Advertising finite commutative semigroups

TL;DR

The paper investigates the structure of finite commutative semigroups by decomposing them into a semilattice of Archimedean components, each of which is a combination of a nilsemigroup and a (possibly trivial) group. It develops and applies tools from closure systems, implications, and generators-and-relations (RFCS) to classify and analyze these components, with a detailed study of cyclic semigroups $C_{m,n}$ and the multiplicative structure $(\mathbb{Z}_n,\odot)$, including the Abelian group of invertible elements $\mathbb{Z}_n^{inv}$. A five-step recipe is presented to extract the global structure from a Cayley table, and the Structure Theorem is established, enabling systematic reconstruction of CF semigroups from their Archimedean parts and idempotent semilattices. The article also develops results on ideal extensions of cyclic semigroups, local confluence, and five open questions to spur further research, tying together semilattices, nilsemigroups, and Abelian groups into a cohesive framework with notable implications for finite ring-like multiplicative structures.

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.