  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.
