Archimedean classes in additive monoids

Summand absorbing submodules are common in modules over (additively) idempotent semirings, for example, in tropical algebra. A submodule $W$ of $V$ is summand absorbing, if $x + y \in W$ implies $x \in W, \; y \in W$ for any $x, y \in V$. This paper proceeds the study of these submodules, and more generally of additive monoids, with emphasis on their archimedean classes and quotient structures.