Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

TL;DR

This paper introduces a new combinatorial approach to the arithmetic of commutative semigroups, proving the Structure Theorem for unions in various non-cancellative cases including ideals and modules, and provides a counterexample.

Contribution

It develops a novel framework for analyzing the structure of sets of lengths in semigroups, extending the theorem to broader classes and identifying a counterexample.

Findings

The Structure Theorem holds for a wide class of semigroups including ideals and modules.

A new characterization of when the Structure Theorem applies is established.

An example of a semigroup that does not satisfy the theorem is provided.

Abstract

Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \in \mathbb N$, let $\mathscr U_k (H)$ denote the set of all $\ell \in \mathbb N$ with the property that there are atoms $u_1, \ldots, u_k, v_1, \ldots, v_{\ell}$ such that $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$ (thus, $\mathscr U_k (H)$ is the union of all sets of lengths containing $k$). The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets $\mathscr U_k (H)$ are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.