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 W implies €x W,y W for any €x,y V. This paper proceeds the study of these submodules, and more generally of additive monoids, with emphasis on their archimedean classes and quotient structures.