Generation of summand absorbing submodules

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.