scholarly journals Amalgamation and extensions of summand absorbing modules over a semiring

2021 ◽  
pp. 2250124
Author(s):  
Zur Izhakian ◽  
Manfred Knebusch
Keyword(s):  
Upper Bound ◽  
Tropical Algebra ◽  
Idempotent Semirings

A submodule [Formula: see text] of [Formula: see text] is summand absorbing, if [Formula: see text] implies [Formula: see text] for any [Formula: see text]. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules.

scholarly journals Generation of summand absorbing submodules

2020 ◽  
pp. 2150201
Author(s):  
Zur Izhakian ◽  
Manfred Knebusch ◽  
Louis Rowen
Keyword(s):  
Krull Dimension ◽  
Sums Of Squares ◽  
Finitely Generated ◽  
Tropical Algebra ◽  
Idempotent Semirings

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.

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