# Generation of summand absorbing submodules

## TL;DR

This paper studies summand absorbing submodules in modules over semirings, providing explicit methods to generate these submodules, which are relevant in tropical algebra and modules lacking zero sums.

## Contribution

It offers an explicit description and generation method for summand absorbing submodules in LZS modules over semirings, extending previous lattice-theoretic analyses.

## Key findings

- Explicit generation of summand absorbing submodules
- Connection to tropical algebra and idempotent semirings
- Extension of lattice-theoretic analysis

## Abstract

An $R$-module $V$ over a semiring $R$ lacks zero sums (LZS) if $ x +y = 0 \; \Rightarrow \; x = y = 0$. More generally, asubmodule $W$ of $V$ is "summand absorbing", if $ \forall \, x, y \in V: \ x + y \in W \; \Rightarrow \; x \in W, \; y \in W. $ These relate to tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this note we describe their explicit generation.