SUBATOMICITY IN RANK-2 LATTICE MONOIDS

Abstract

Let $M$ be a cancellative and commutative monoid (written additively). The monoid $M$ is atomic if every noninvertible 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, quasiatomic, 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.