Abstract
An R-module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, we call a submodule W of V "summand absorbing" (SA) in V 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 arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of ...
Abstract
An R-module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, we call a submodule W of V "summand absorbing" (SA) in V 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 arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan-Holder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation. (C) 2018 Elsevier B.V. All rights reserved.
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