Subatomicity in Rank-2 Lattice Monoids

TL;DR

This paper explores the atomic properties of lattice monoids, focusing on various weaker forms of atomicity such as nearly atomic, almost atomic, quasi-atomic, and Furstenberg, to understand their structure.

Contribution

It provides a detailed analysis of the atomic structure of lattice monoids, emphasizing the behavior of weaker atomic properties within this class.

Findings

Characterization of nearly atomic and almost atomic lattice monoids

Identification of conditions for quasi-atomic and Furstenberg properties

Insights into the atomic decomposition in submonoids of free abelian groups

Abstract

Let $M$ be a cancellative and commutative monoid (written additively). The monoid $M$ is atomic if every non-invertible element can be written as a sum of irreducible elements (often called atoms in the literature). Weaker versions of atomicity have been recently introduced and investigated, including the properties of being nearly atomic, almost atomic, quasi-atomic, and Furstenberg. In this paper, we investigate the atomic structure of lattice monoids, (i.e., submonoids of a finite-rank free abelian group), putting special emphasis on the four mentioned atomic properties.