Generation of summand absorbing submodules

Abstract

An [Formula: see text]-module [Formula: see text] over a semiring [Formula: see text] lacks zero sums (LZS) if [Formula: see text] implies [Formula: see text]. More generally, a submodule [Formula: see text] of [Formula: see text] is “summand absorbing” (SA), if, for all [Formula: see text], [Formula: see text] These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation.