Rees quotients of numerical semigroups

TL;DR

This paper introduces a new class of finite semigroups derived from Rees quotients of numerical semigroups, explores their properties, and establishes their role as generators for certain pseudovarieties.

Contribution

It defines and analyzes Rees quotients of numerical semigroups, providing presentations and identifying generators for the pseudovariety of commutative and nilpotent semigroups.

Findings

Rees quotients of N generate the pseudovariety of commutative, nilpotent semigroups.

Presented explicit semigroup presentations.

Answered several natural questions about these semigroups.

Abstract

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of N, the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.