Abstract
An R-module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, a submodule W of V is "summand absorbing" (SA), if, for all x,y is an element of V, x + y is an element of W double right arrow x is an element of W,y is an element of W. These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums ...
Abstract
An R-module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, a submodule W of V is "summand absorbing" (SA), if, for all x,y is an element of V, x + y is an element of W double right arrow x is an element of W,y is an element of W. 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.
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