Archimedean classes in additive monoids

TL;DR

This paper studies the structure of summand absorbing submodules and additive monoids, focusing on their archimedean classes and quotient structures, with applications in tropical algebra.

Contribution

It advances the understanding of archimedean classes in additive monoids and explores quotient structures, extending prior work on summand absorbing submodules.

Findings

Characterization of archimedean classes in additive monoids

Analysis of quotient structures of these monoids

Application insights for tropical algebra

Abstract

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.